Reuman's dimension conjecture for shrunken affine Deligne–Lusztig varieties

Let $G$ be a reductive group with extended affine Weyl group $\widetilde W$, finite Weyl group $W$, and simple reflections $S$. Let $[b]$ be a basic $\sigma$-conjugacy class. For $x\in\widetilde W$, let $\eta_1(x)$ be its finite Weyl-group projection and let $\eta_2(x)$ be the Weyl chamber containing the alcove $x\mathbf a$. Let $\eta_G$ be the Kottwitz invariant, let $W_T$ be the subgroup generated by $T\subseteq S$, let $\ell$ be the length function, and let $\operatorname{def}_G(b)$ be the defect.

Reuman's conjecture. If $x$ lies in the shrunken Weyl chambers, then $X_x(b)\ne\emptyset$ if and only if

$$\eta_G(x)=\eta_G(b),\qquad \eta_2(x)^{-1}\eta_1(x)\eta_2(x)\in W\setminus\bigcup_{T\subsetneq S}W_T.$$

When these conditions hold,

$$\dim X_x(b)=\frac{1}{2}\left(\ell(x)+\ell\left(\eta_2(x)^{-1}\eta_1(x)\eta_2(x)\right)-\operatorname{def}_G(b)\right).$$

This extends the earlier conjecture for $b=1$ to all basic $b$; the paper reports computer evidence but does not prove the full assertion.