The characterization of pairs occurring infinitely often among slowest walks

Let $T$ be a set of relatively prime pairs. For $(\alpha,\beta)\in T$, let $s^{\alpha,\beta}(n)$ denote the corresponding slow-walk function, define

$$\mathbf{s}_T(n)=\max_{(\alpha,\beta)\in T}s^{\alpha,\beta}(n),\qquad \mathbf{S}_T(n)=\{(\alpha,\beta):s^{\alpha,\beta}(n)=\mathbf{s}_T(n)\},$$

and let

$$\Gamma=\Gamma_T=\min\{\gamma_{\alpha,\beta}:(\alpha,\beta)\in T\},\qquad R_T'=\{(\alpha,\beta)\in T:\gamma_{\alpha,\beta}<\Gamma^2\}.$$

Characterization conjecture. There exist infinitely many $n$ with $(\alpha,\beta)\in\mathbf{S}_T(n)$ if and only if $(\alpha,\beta)\in R_T'$. This concerns which relatively prime parameter pairs occur infinitely often as maximizers of the slow-walk function; the paper notes that the only-if direction may require handling the boundary case $\gamma_{\alpha,\beta}=\Gamma^2$, while the if direction is left open.