The P-alcove conjecture for basic affine Deligne–Lusztig varieties

Let $G$ be a reductive group over the relevant local field, let $L$ be its maximal unramified extension, and let $X_x(b)$ denote the affine Deligne–Lusztig variety in the affine flag variety attached to $x ewlinein \widetilde W$ and $b\in G(L)$. Let $[b]$ be a basic $\sigma$-conjugacy class. For a semistandard parabolic subgroup $P=MN$, call $x\mathbf a$ a $P$-alcove when it satisfies the corresponding $P$-alcove condition, and let $\eta_M:M(L)\to\Lambda_M$ be the Kottwitz homomorphism.

P-alcove conjecture. $X_x(b)\ne\emptyset$ if and only if, for every semistandard $P=MN$ for which $x\mathbf a$ is a $P$-alcove, $b$ is $\sigma$-conjugate to an element $b'\in M(L)$ and $x$ and $b'$ have the same image under $\eta_M$.

One direction, namely the predicted emptiness, is proved in the paper; the converse, asserting non-emptiness whenever the conditions hold, remains open. For shrunken $x$, the conjecture reduces to Reuman's conjecture.