DeBacker's homogeneity conjecture for invariant distributions

Let $F$ be a local non-Archimedean field, let $G=\operatorname{GL}(n,F)$ with Lie algebra $\mathfrak{g}$, and let $\mathcal B$ be its Bruhat–Tits building. Fix an irreducible smooth representation $(\pi,V)$ of $G$ with depth $\rho(\pi)$, choose $r$ such that $\mathfrak{g}_r\ne\mathfrak{g}_{r^+}=\mathfrak{g}_{-\rho(\pi)}$, and define $D_{r^+}$, $\widetilde J_{r^+}$, and $J(\mathcal N)$ as in the source. Here $\mathcal N$ is the nilpotent cone, $J(\mathcal N)$ is the space of invariant distributions supported on it, and $\operatorname{res}_{D_{r^+}}$ denotes restriction to $D_{r^+}$. DeBacker's homogeneity conjecture. One has

$$\operatorname{res}_{D_{r^+}}\widetilde J_{r^+}=\operatorname{res}_{D_{r^+}}J(\mathcal N).$$

The text states that a theorem on homogeneity follows from this conjecture, but supplies no resolution of the conjecture itself. It is therefore recorded as open.