The shrunken Weyl chamber conjecture for affine Deligne–Lusztig varieties

Let $G$ be the reductive group in the setup, let $x=w\varepsilon^\mu\in\widetilde W$ lie in a shrunken Weyl chamber, so that $\operatorname{LP}(x)=\{v\}$ for a uniquely determined $v\in W$, and let $[b]\in B(G)$. If $X_x(b)\neq\emptyset$, define $D\in\mathbb Z_{\geq 0}$ by

$$\dim X_x(b)=\frac{1}{2}\left(\ell(x)+D-\langle\nu(b),2\rho\rangle-\operatorname{defect}(b)\right),$$

and let $C\in\mathbb Z_{\geq 1}$ be the number of $J_b(F)$-orbits of top-dimensional irreducible components of $X_x(b)$. Write $\lambda(b)$ for the relevant Kottwitz invariant and define either of the multisets

$$E_1=\{e\mid(\omega,e)\in\operatorname{wts}(v\Rightarrow\sigma(wv)),\ \lambda(b)\equiv v^{-1}\mu-\omega\in X_\ast(T)_\Gamma\}_m,$$ $$E_2=\{e\mid(\omega,e)\in\operatorname{wts}(v\Rightarrow\sigma(wv)),\ \nu(b)=\operatorname{conv}(v^{-1}\mu-\omega)\}_m.$$

Let $E$ be either $E_1$ or $E_2$. The shrunken Weyl chamber conjecture. The predictions are: (a) $X_x(b)\neq\emptyset$ if and only if $E\neq\emptyset$ and $\kappa(x)=\kappa(b)\in\pi_1(G)_\Gamma$; for $E=E_1$, the Kottwitz condition is automatic. (b) If $X_x(b)\neq\emptyset$, then $\max(E)=D$. (c) If $X_x(b)\neq\emptyset$, then $C$ is at most the multiplicity of $D$ in $E$, which may be $+\infty$ for $E_2$.