Equivalence of conditions $(T)|l$ and $(T')|l$ for uniformly elliptic RWRE

Consider a random walk in a uniformly elliptic random environment in dimension $d\ge 2$ and a direction $l\in\mathbb S^{d-1}$. For $\gamma\in(0,1]$, condition $(T)_\gamma|l$ requires

$$\limsup_{L\to\infty}L^\gamma\log P_0(\tilde T_{-L}^{l'}<T_L^{l'})<0$$

for every $l'$ in a neighborhood of $l$; condition $(T')|l$ requires $(T)_\gamma|l$ for every $\gamma\in(0,1)$, while $(T)|l$ means $(T)_1|l$. $(T)|l$–$(T')|l$ equivalence conjecture. Condition $(T)|l$ is equivalent to condition $(T')|l$. Results cited in the source establish several implications and equivalences for intermediate exponents, but this equivalence is explicitly presented as still open.