Stable-center conjecture for the depth-$r$ image

Assume that the residue characteristic of $k$ is sufficiently large. For $r\in\mathbb{Z}_{(p)}\cap\mathbb{Q}_{\geq 0}$, let $\xi^r$ be the map constructed in the paper, and let $\mathcal{Z}^{st}(G)$ and $\mathcal{Z}^{r}(G)$ denote the stable and depth-$r$ parts of the Bernstein center. Set

$$\mathcal{Z}^{st,r}(G)=\mathcal{Z}^{st}(G)\cap\mathcal{Z}^{r}(G).$$

Depth-$r$ stable-center conjecture. The image of $\xi^r$ is contained in the depth-$r$ stable center:

$$\operatorname{Im}(\xi^r)\subset\mathcal{Z}^{st,r}(G).$$

This predicts that the depth-$r$ central elements constructed in the paper act through stable distributions. It is presented as a consequence suggested by the conjectural description of the stable center and related Langlands-parameter conjectures.