The Weyl-group bound for components of affine Springer fibers

Let $\mathfrak{g}$ be the Lie algebra of $G$ over the local field $F$, let $\mathcal{X}_\gamma$ be the affine Springer fiber associated with $\gamma$, and let $\mathcal{T}$ act on its geometric base change $\mathcal{X}_\gamma\times\operatorname{Spec}\bar{k}$. Let $W$ be the Weyl group, and let $\operatorname{depth}(\gamma)$ denote the depth of $\gamma$.

Weyl-group component bound conjecture. For any regular semisimple $\gamma\in\mathfrak{g}$, the number of $\mathcal{T}$-orbits of components of $\mathcal{X}_\gamma\times\operatorname{Spec}\bar{k}$ is at most the order of the Weyl group. Moreover, equality holds when

$$\operatorname{depth}(\gamma)=1.$$

This conjecture concerns the expected Weyl-group-sized control of geometric components of affine Springer fibers and is stated alongside the regular-representation conjecture. The source gives no evidence of resolution.