Eventual translation-shift conjecture for affine Deligne–Lusztig varieties

Let $G$ be one of the groups $SL_2$, $SL_3$, or $Sp_4$. Let $b=\epsilon^\nu$, and let $\ell(b)$ denote the length of the translation in $\widetilde W$ determined by $\nu$. Eventual translation-shift conjecture. There exists $n_0$ such that, for every $x\in\widetilde W$ with $\ell(x)\ge n_0$,

$$X_x(b)\ne\emptyset \quad\Longleftrightarrow\quad X_x(1)\ne\emptyset.$$

When these varieties are non-empty, their dimensions satisfy

$$\dim X_x(b)=\dim X_x(1)-\frac{1}{2}\ell(b).$$

The conjecture concerns the asymptotic behavior of affine Deligne–Lusztig varieties for the three specified groups and nontrivial $b$. The source states it as a conjecture and provides no resolution status.