A subsequence conjecture for pairs in $R_T'$

Let $T$ be a set of relatively prime pairs, and let $(\alpha,\beta)\in R_T'$, where

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

Let $g_t^{\alpha,\beta}$ denote the sequence associated with the $(\alpha,\beta)$-walk, and let $\mathbf{S}_T(n)$ be the set of pairs attaining $\mathbf{s}_T(n)=\max_{(\alpha,\beta)\in T}s^{\alpha,\beta}(n)$. Subsequence conjecture. There exist infinitely many $t$ such that

$$(\alpha,\beta)\in\mathbf{S}_T\bigl(g_t^{\alpha,\beta}+\beta g_{t-1}^{\alpha,\beta}\bigr).$$

This is proposed as one possible route to proving the if direction of the preceding characterization conjecture, and it remains open.