The affine Springer fiber orbital-integral boundedness conjecture

Let $F$ be the local field, let $\mathfrak{g}$ be the Lie algebra of the reductive group $G$ over $F$, and fix an alcove $x$ with $\mathfrak{g}_{x,\geq 0}$ its associated Moy–Prasad lattice. For $\gamma\in\mathfrak{g}$, write $I_\gamma(1_{\mathfrak{g}_{x,\geq 0}})$ for the orbital integral of the characteristic function of this lattice.

Orbital-integral boundedness conjecture. For any $\gamma\in\mathfrak{g}$,

$$I_\gamma(1_{\mathfrak{g}_{x,\geq 0}})=O(1).$$

This would give a uniform boundedness statement for orbital integrals, extending the preceding result for regular semisimple elements and the proposition for nilpotent elements. The source presents it as a conjecture and gives no resolution.