here the implicit constants here depend on $K$, $C_1$, $s$, and $\beta$ (but not $T$): $$\begin{aligned}
 {N_{\text{I}}}^{1-} {N}^{-2-} \gtrsim T \quad &\Leftrightarrow \quad {N_{\text{I}}}\gtrsim T^{\frac{1}{2s-1}+} \text{\quad\&\quad} {N_{\text{I}}}\gtrsim  T^{1+} {N_{\text{HL}}}^{1 - 2\beta    +}, 
\\  {N_{\text{I}}}^{2-} {N}^{-4-} \gtrsim T \quad &\Leftrightarrow \quad {N_{\text{I}}}\gtrsim  T^{\frac{1}{4s-2}+} \text{\quad\&\quad} {N_{\text{I}}}\gtrsim  T^{\frac12+}  {N_{\text{HL}}}^{1 - 2\beta    +}, \\
 {N_{\text{I}}}^{\frac32 -} {N}^{-3-} \gtrsim T \quad &\Leftrightarrow \quad {N_{\text{I}}}\gtrsim  T^{\frac{2}{6s-3}+} \text{\quad\&\quad} {N_{\text{I}}}\gtrsim  T^{\frac23+}  {N_{\text{HL}}}^{1 - 2\beta    +}.
 \end{aligned}$$ The final term in the minimum yields the most complicated constraint: $$\begin{aligned}
 {N_{\text{I}}}^{-(1-s)} {N_{\text{HL}}}^{-(s-\frac12 -\beta   )} {N}^{-1-} \gtrsim T \quad &\Leftrightarrow \quad {N_{\text{I}}}\lesssim  T^{\frac{-1+}{2(1-s)}}{N_{\text{HL}}}^{\frac{\beta + \frac12 - s}{2(1-s)}-} \text{\quad\&\quad} {N_{\text{I}}}\lesssim  T^{\frac{-1}{1-s}}  {N_{\text{HL}}}^{\frac{2\beta - s }{1-s}-}. 
 \end{aligned}$$

To satisfy all these constraints simultaneously, we require $$s > \frac12$$ and take $${N_{\text{I}}}= {N_{\text{HL}}}^\alpha \text{ for some } \alpha \in \Biggl( 1 - 2\beta   , \min\Bigl\{\frac{\beta + \frac12 - s }{2(1-s)}, \; \frac{2\beta - s   }{1-s}  \Bigr\} \Biggr)$$ and some ${N_{\text{HL}}}$ very large, dependent on $K$, $T$, $s$, and $r$. This is possible as long as $$1- 2\beta    <  \min\Bigl\{\frac{\beta + \frac12 - s }{2(1-s)}, \; \frac{2\beta - s   }{1-s}  \Bigr\}.$$ Solving this with $s > \frac12$, we find the constraint $$\beta > \frac12 - \min\Bigl\{\frac{1-s}{4(1-s) +1}, \; \frac{1-s}{2(1-s) +2} \Bigr\}    = \frac12 -  \frac{1-s}{2(1-s) 