$J$-relative Cuntz-Pimsner algebras, for $J$ inside Katsura’s ideal (Theorem \[T: sme\]). We highlight that the converse of the rigidity question has been exhibited to be considerably difficult even when ${\stackrel{\textup{SME}}{\thicksim}}$ is substituted by honest isomorphisms. In this reduced form, it is better known as the Conjugacy Problem and it has been answered in several major classes of C\*-correspondences, e.g. [@DavKak12; @DavKat08; @DavKat11; @DavRoy11; @Dor15; @KatKri04; @KakSha15; @Sol04] to mention but a few.

We derive our motivation also from further researching ${\stackrel{\textup{SME}}{\thicksim}}$ and ${\, \stackrel{\textup{$\Delta$}}{\thicksim} \,}$. The differences between these relations are subtle and it is natural to ask when they actually coincide. We show that this is true for tensor algebras of aperiodic non-degenerate C\*-correspondences. In particular we prove that $E {\stackrel{\textup{SME}}{\thicksim}}F$ if and only if ${{\mathcal{T}}}_E^+ {\, \stackrel{\textup{$\Delta$}}{\thicksim} \,}{{\mathcal{T}}}_F^+$ if and only if ${{\mathcal{T}}}_E^+ {\stackrel{\textup{SME}}{\thicksim}}{{\mathcal{T}}}_F^+$ (Corollary \[C: aperiodic\]). This is quite pleasing as we incorporate a big class of operator algebras with approximate identities. Recall that ${\stackrel{\textup{SME}}{\thicksim}}$ and ${\, \stackrel{\textup{$\Delta$}}{\thicksim} \,}$ are shown to be different even for unital operator algebras.

Our results read the same if ${\stackrel{\textup{SME}}{\thicksim}}$ is substituted by stable isomorphisms (Corollary \[C: stable\]). This follows directly from our analysis and the observation that stable isomorphism coincides 