Sznitman's quenched atypical-exit estimate conjecture

Let $d\geq2$, let $\mathbb{P}$ be uniformly elliptic, and let $l\in\mathbb{S}^{d-1}$. Assume condition $(T')|l$ holds. For $\beta\in(0,1)$ and $L>0$, define the slab

$$U_{\beta,l,L}:=\{x\in\mathbb{Z}^d:-L^\beta\leq x\cdot l\leq L\}.$$

Let $T_B:=\inf\{n\in\mathbb{N}_0:X_n\in B\}$ and write $T_{\partial U_{\beta,l,L}}$ for the first hitting time of the boundary of this slab. Sznitman's quenched exit estimate conjecture. For every $c>0$, every $\beta\in(0,1)$, and every $\alpha\in(0,\beta d)$,

$$\limsup_{L\to\infty}L^{-\alpha}\log\mathbb{P}\left(P_{0,\omega}\left(X_{T_{\partial U_{\beta,l,L}}}\cdot l>0\right)\leq e^{-cL^\beta}\right)<0.$$

The conjecture predicts stretched-exponential decay, under the environment law, of the probability that the quenched walk has an atypically small chance of exiting the slab through its forward side. It was conjectured by Sznitman and is presented in the source as open; the surrounding paper proves related equivalences between ballisticity conditions in dimensions at least four.