He’s conjecture on connected components of affine Deligne–Lusztig varieties

Let $G$ be the reductive group, let $b\in G(\breve{F})$ and let $\mu$ be a cocharacter, with associated affine Deligne–Lusztig variety $X^{\mathcal{K}_p}_\mu(b)$. Write $\pi_0(X^{\mathcal{K}_p}_\mu(b))$ for its set of connected components, and let $\omega_G$ be the map to the subset $c_{b,\mu}\pi_1(G)_I^\varphi$ of the coinvariants of the algebraic fundamental group $\pi_1(G)$ under inertia $I$ and Frobenius $\varphi$. Assume that $(b,\mu)$ is HN-irreducible. He’s conjecture. The map

$$\omega_G:\pi_0(X^{\mathcal{K}_p}_\mu(b))\to c_{b,\mu}\pi_1(G)_I^\varphi$$

is bijective. The connected components of affine Deligne–Lusztig varieties are an important part of understanding their geometric structure. The statement is presented as a conjecture suggested by X. He; its resolution is not specified in the supplied text.