user1485: I am trying to learn single rook endgames with two pawns against one, with added stipulation that the two pawns of the attacking side are on neighboring files with the pawn of the defending side also on one of those files. It could for instance be e+f pawns vs f pawn or g+h pawns vs g pawn.
What is the most complicated such position that is still won for the attacking side? To make the question precise, I am asking for the position with the maximum depth to conversion.

user9423: Download the Lomonosov endgame Android application, it is what you are looking for. This probably solves your issue if you toy around for a while.

In the Play Store, it looks like this.

Hope this helps!

user3122: Just this week the 7-piece Syzygy table for this endgame was published and is now available online, together with some basic statistics about this endgame. It gives a maximum DTZ of 136 (in a position that happens to match your criteria).
So in any case 136 is a lower bound for the maximum DTC.
To get from DTZ to DTC for paths leading through this position we just need to check: (1) It is indeed a conversion (by capturing the black pawn), instead of just a pawn push. (2) The phase was not entered by a non-capturing pawn push.
For (1) see the DTZ-mainline:
[FEN "6k1/8/3pP1R1/3P4/8/1K6/8/3r4 b - - 0 1"]

1... Kf8 2. Rf6+ Ke8 3. Kc4 Rc1+ 4. Kd3 Rd1+ 5. Ke4
Re1+ 6. Kf4 Rf1+ 7. Kg5 Rg1+ 8. Kh6 Rh1+ 9. Kg7
Rg1+ 10. Rg6 Rd1 11. Rg5 Ke7 12. Rf5 Rg1+ 13. Kh6
Rd1 14. Kg5 Rg1+ 15. Kf4 Rf1+ 16. Ke4 Re1+ 17. Kd3
Rd1+ 18. Ke3 Ke8 19. Ke4 Re1+ 20. Kf4 Rg1 21. Rh5
Rd1 22. Ke4 Re1+ 23. Kf5 Rf1+ 24. Kg5 Rg1+ 25. Kf6
Rf1+ 26. Rf5 Rd1 27. Rf2 Rd4 28. Ra2 Rf4+ 29. Kg5
Rd4 30. Ra1 Rd2 31. Kf5 Rd4 32. Ra7 Rd1 33. Ra4
Ke7 34. Ke4 Re1+ 35. Kd3 Rd1+ 36. Kc4 Rc1+ 37. Kb4
Rb1+ 38. Kc3 Rc1+ 39. Kd2 Rc5 40. Rd4 Ra5 41. Ke3
Ra3+ 42. Rd3 Ra1 43. Rb3 Re1+ 44. Kd2 Re5 45. Rb5
Rg5 46. Kc3 Kf6 47. Kb4 Rg1 48. Ka5 Ra1+ 49. Kb6
Ke7 50. Ra5 Rb1+ 51. Kc6 Rc1+ 52. Kb5 Rb1+ 53. Kc4
Rc1+ 54. Kd3 Rd1+ 55. Kc3 Rc1+ 56. Kd2 Rc8 57. Kd3
Rc7 58. Ra1 Rb7 59. Rh1 Rb5 60. Rh7+ Kf6 61. Rf7+
Kg6 62. Kc4 Rc5+ 63. Kd4 Ra5 64. Rc7 Ra1 65. Rc6
Rd1+ 66. Kc4 Rc1+ 67. Kb3 Rb1+ 68. Kc2 Ra1 69. Rxd6

For (2) it helps that white is giving check, so the last move was:

A capture on g6. But that would have started the phase.
Rf6-g6+. But e7 would have been better.
Rh6-g6+. But Rf6 would have been better.

Admittedly it might still be possible that there are entirely different positions, stringed together by pawn pushes, that achieve a higher DTC (total DTZ).

user8860: As Glorfindel notes,
the Lomonosov tables are great but might not have the answer you want.
In RPP/RP, according to the list in
this
article by Guy Haworth, we find that the maximal distance to mate
is 200+ moves, from the following position -- which happens not to have
the White pawns connected, but worse for your purpose,
it looks like most of the play is post-conversion:
[Title "Black to move; White eventually wins"]
[Fen "8/k1P5/8/8/1R5P/7p/r7/1K6 b - - 0 1"]
[StartFlipped "0"]

1...h2 2.Kxa2 h1=Q 3.c8=R!

after Kxa2 we're in 6-man territory, and the k4it.de site says
"Win in 216" starting Qd5+ 4 Kb1 Qa5(d6) -- that's right, not Qd1(h1)+,
which lets White win in only 200, nor Qd3+, which hastens the end
by another move (199).
Possibly Guy Haworth has enough access to 7-man tables
to get the answer to your query.

1485: <p>I am trying to learn single rook endgames with two pawns against one, with added stipulation that the two pawns of the attacking side are on neighboring files with the pawn of the defending side also on one of those files. It could for instance be <code>e</code>+<code>f</code> pawns vs <code>f</code> pawn or <code>g</code>+<code>h</code> pawns vs <code>g</code> pawn.</p>

<p>What is the most complicated such position that is still won for the attacking side? To make the question precise, I am asking for the position with the maximum <a href="https://en.wikipedia.org/wiki/Endgame_tablebase#Metrics:_Depth_to_conversion_and_depth_to_mate" rel="noreferrer">depth to conversion</a>.</p>
 9423: <p>Download the <a href="https://play.google.com/store/apps/details?id=com.convekta.android.lomonosovtb" rel="nofollow noreferrer">Lomonosov endgame Android application</a>, it is what you are looking for. This probably solves your issue if you toy around for a while.</p>

<p><a href="https://i.stack.imgur.com/ZOXjEl.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZOXjEl.jpg" alt="enter image description here"></a></p>

<p>In the Play Store, it looks like this.</p>

<p><a href="https://i.stack.imgur.com/bGKmtl.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bGKmtl.jpg" alt="enter image description here"></a></p>

<p>Hope this helps!</p>
 3122: <p>Just this week the 7-piece Syzygy table for this endgame was published and is now available online, together with some basic statistics about this endgame. It <a href="https://syzygy-tables.info/?fen=6k1/8/3pP1R1/3P4/8/1K6/8/3r4_b_-_-_0_1" rel="nofollow noreferrer">gives</a> a maximum DTZ of 136 (in a position that happens to match your criteria).</p>

<p>So in any case 136 is a lower bound for the maximum DTC.</p>

<p>To get from DTZ to DTC for paths leading through this position we just need to check: (1) It is indeed a conversion (by capturing the black pawn), instead of just a pawn push. (2) The phase was not entered by a non-capturing pawn push.</p>

<p>For (1) see the DTZ-mainline:</p>

<pre><code>[FEN "6k1/8/3pP1R1/3P4/8/1K6/8/3r4 b - - 0 1"]

1... Kf8 2. Rf6+ Ke8 3. Kc4 Rc1+ 4. Kd3 Rd1+ 5. Ke4
Re1+ 6. Kf4 Rf1+ 7. Kg5 Rg1+ 8. Kh6 Rh1+ 9. Kg7
Rg1+ 10. Rg6 Rd1 11. Rg5 Ke7 12. Rf5 Rg1+ 13. Kh6
Rd1 14. Kg5 Rg1+ 15. Kf4 Rf1+ 16. Ke4 Re1+ 17. Kd3
Rd1+ 18. Ke3 Ke8 19. Ke4 Re1+ 20. Kf4 Rg1 21. Rh5
Rd1 22. Ke4 Re1+ 23. Kf5 Rf1+ 24. Kg5 Rg1+ 25. Kf6
Rf1+ 26. Rf5 Rd1 27. Rf2 Rd4 28. Ra2 Rf4+ 29. Kg5
Rd4 30. Ra1 Rd2 31. Kf5 Rd4 32. Ra7 Rd1 33. Ra4
Ke7 34. Ke4 Re1+ 35. Kd3 Rd1+ 36. Kc4 Rc1+ 37. Kb4
Rb1+ 38. Kc3 Rc1+ 39. Kd2 Rc5 40. Rd4 Ra5 41. Ke3
Ra3+ 42. Rd3 Ra1 43. Rb3 Re1+ 44. Kd2 Re5 45. Rb5
Rg5 46. Kc3 Kf6 47. Kb4 Rg1 48. Ka5 Ra1+ 49. Kb6
Ke7 50. Ra5 Rb1+ 51. Kc6 Rc1+ 52. Kb5 Rb1+ 53. Kc4
Rc1+ 54. Kd3 Rd1+ 55. Kc3 Rc1+ 56. Kd2 Rc8 57. Kd3
Rc7 58. Ra1 Rb7 59. Rh1 Rb5 60. Rh7+ Kf6 61. Rf7+
Kg6 62. Kc4 Rc5+ 63. Kd4 Ra5 64. Rc7 Ra1 65. Rc6
Rd1+ 66. Kc4 Rc1+ 67. Kb3 Rb1+ 68. Kc2 Ra1 69. Rxd6
</code></pre>

<p>For (2) it helps that white is giving check, so the last move was:</p>

<ul>
<li>A capture on g6. But that would have started the phase.</li>
<li>Rf6-g6+. But e7 would have been better.</li>
<li>Rh6-g6+. But Rf6 would have been better.</li>
</ul>

<p>Admittedly it might still be possible that there are entirely different positions, stringed together by pawn pushes, that achieve a higher DTC (total DTZ).</p>
 8860: <p>As <strong>Glorfindel</strong> notes,
the Lomonosov tables are great but might not have the answer you want.
In RPP/RP, according to the list in
<a href="http://centaur.reading.ac.uk/34268/2/chessendrecann.pdf" rel="nofollow noreferrer">this
article</a> by Guy Haworth, we find that the maximal distance to mate
is 200+ moves, from the following position -- which happens not to have
the White pawns connected, but worse for your purpose,
it looks like most of the play is post-conversion:</p>

<pre><code>[Title "Black to move; White eventually wins"]
[Fen "8/k1P5/8/8/1R5P/7p/r7/1K6 b - - 0 1"]
[StartFlipped "0"]

1...h2 2.Kxa2 h1=Q 3.c8=R!
</code></pre>

<p>after Kxa2 we're in 6-man territory, and the k4it.de site says
"Win in 216" starting Qd5+ 4 Kb1 Qa5(d6) -- that's right, not Qd1(h1)+,
which lets White win in only 200, nor Qd3+, which hastens the end
by another move (199).</p>

<p>Possibly Guy Haworth has enough access to 7-man tables
to get the answer to your query.</p>
