Chen–Zhu's conjecture on irreducible components of affine Deligne–Lusztig varieties

Let $G$ be the reductive group in the local Shimura-variety setting, let $mu$ be the relevant dominant cocharacter, and let $b\in B(G)$ be a $\sigma$-conjugacy class. Write $X_\mu(b)$ for the corresponding affine Deligne–Lusztig variety, $J_b(F)$ for the group acting on it, and let $la_b\in\mathbb{X}^\bullet(\hat T)_\sigma$ be the element attached to $b$, described as the best integral approximation to the Newton point $nu_b$. Chen–Zhu's conjecture. The $J_b(F)$-orbits of the set of irreducible components of $X_\mu(b)$ canonically give a basis of

$$(V_\mu|_{\hat G^\sigma})(\lambda_b).$$

This conjecture predicts a representation-theoretic description of the irreducible components of affine Deligne–Lusztig varieties and extends the finite Deligne–Lusztig setting. The source gives no resolution.