The basic-class comparison conjecture for affine Deligne–Lusztig varieties

Let $G$ be a reductive group with extended affine Weyl group $\widetilde W$, and let $X_x(b)$ be the associated affine Deligne–Lusztig variety. For an arbitrary $\sigma$-conjugacy class $[b]$, let $[b_{\rm b}]$ be the unique basic $\sigma$-conjugacy class with $\eta_G(b)=\eta_G(b_{\rm b})$, let $\nu$ be the Newton point of $b$, and let $\rho$ be the half-sum of positive roots.

Basic-class comparison conjecture. There exists $N_b\in\mathbb Z_{\ge 0}$ such that, for every $x\in\widetilde W$ with $\ell(x)\ge N_b$,

$$X_x(b)\ne\emptyset\Longleftrightarrow X_x(b_{\rm b})\ne\emptyset.$$

When these varieties are non-empty,

$$\dim X_x(b)=\dim X_x(b_{\rm b})-\frac{1}{2}\left(\langle 2\rho,\nu\rangle+\operatorname{def}_G(b)-\operatorname{def}_G(b_{\rm b})\right).$$

This extends the dimension conjecture for translation elements to arbitrary $b$ and sufficiently long $x$. The paper presents it as conjectural, supported by computer calculations.