The integral basic-element conjecture for affine Deligne–Lusztig varieties

Let $x\in\widetilde W$ and let $[b]\in B(G)$ be integral, meaning that $\operatorname{defect}(b)=0$. For each $v\in\operatorname{LP}(x)$ and $u\in W$, define the multiset

$$E(u,v)=\{e\mid(\omega,e)\in\operatorname{wts}(u\Rightarrow\sigma(wu)\dashrightarrow\sigma(wv)),\ u^{-1}\mu-\omega=\lambda(b)\in X_\ast(T)_\Gamma\}_m.$$

Set $\max\emptyset=-\infty$, and define

$$d=\max_{u\in W}\min_{v\in\operatorname{LP}(x)}\max(E(u,v))\in\mathbb Z_{\geq0}\cup\{-\infty\},$$ $$c=\sum_{u\in W}\min_{v\in W}\bigl(\text{multiplicity of }d\text{ in }E(u,v)\bigr)\in\mathbb Z_{\geq0}.$$

Let $D$ and $C$ be defined by the dimension formula and component-orbit count above. The integral basic-element conjecture. The predictions are: (a) if for every $u\in W$ there is some $v\in\operatorname{LP}(x)$ with $E(u,v)=\emptyset$, equivalently $d=-\infty$, then $X_x(b)=\emptyset$; (b) if $X_x(b)\neq\emptyset$, then $D\leq d$; (c) if $X_x(b)\neq\emptyset$ and $D=d$, then $C\leq c$; (d) if $[b]$ satisfies $\langle\nu(b),\alpha\rangle\geq1$ for every $\alpha\in\Phi^+$, then $X_x(b)\neq\emptyset$ and $D=d$.