Connected-component bijection for Hodge–Newton irreducible affine Deligne–Lusztig varieties

Let $G$ be the reductive group, $L$ its relevant local field extension, $P$ the parahoric subgroup, $\lambda$ a coweight, and $b\in G(L)$. Let $X(\lambda,b)_P$ be the associated closed affine Deligne–Lusztig variety, let $\pi_0(X(\lambda,b)_P)$ denote its set of connected components, and let

$$\eta_G:G(L)/P\to\pi_1(G)$$

be the natural projection. Assume that $(\lambda,b)$ is Hodge–Newton irreducible. Connected-component bijection. The map $\eta_G$ induces a bijection

$$\pi_0(X(\lambda,b)_P)\cong (\sigma-1)^{-1}(\eta_G(t^\lambda)-\eta_G(b))\subseteq\pi_1(G).$$

This gives an explicit description of the connected components in the Hodge–Newton irreducible case, completing the third step of the paper’s reduction to adjoint simple groups and Hodge–Newton indecomposable pairs. The supplied text does not state whether the claim has been proved or remains conjectural.