Stability conjecture for the depth-$r$ Bernstein-center image

Let $G$ be the group under consideration, let $r$ be an integral depth, and let $\mathcal Z^{st,r}(G)=\mathcal Z^r(G)\cap\mathcal Z^{st}(G)$ denote the subspace of stable elements in the depth-$r$ Bernstein center. Let

$$\xi^r:C^{st}(P_r/P_r^+)\to\mathcal Z^r(G)$$

be the algebra homomorphism constructed in the preceding theorem. Stability conjecture. We have

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

The image consists of elements constructed from stable functions on the finite reductive group or Lie algebra and is expected to retain stability in the Bernstein center. The statement has been proved: the element $z^r=\xi^r(\mathrm 1_e)$ is the depth-$r$ Bernstein projector, whose stability was established in the cited result, although the displayed image-level formulation is the conjectural statement recorded here.