The Euler-to-homological convolution equivalence conjecture

Let $\mathrm{\mathbf{econv}}(\mathrm{Gr})_0$ be the subcategory of constructible-function convolutions generated by the relevant Satake-fibre functions, and let $\mathrm{\mathbf{hconv}}(\mathrm{Gr})$ be the corresponding category of top-homology convolutions. For a constructible function $f$ on a convolution variety $Z$, define

$$\Xi(f)=\sum_{X\in\operatorname{Irr}(Z)}f(X)[X],$$

where $f(X)$ is the value of $f$ on a dense open subset of the irreducible variety $X$. Euler-to-homological convolution conjecture. The map $\Xi$ between hom spaces restricts to an equivalence of pivotal categories from $\mathrm{\mathbf{econv}}(\mathrm{Gr})_0$ to $\mathrm{\mathbf{hconv}}(\mathrm{Gr})$, up to a sign correction of the tensor and pivotal structures. The conjecture would connect the Euler-characteristic and homological versions of geometric Satake and would imply the earlier representation-category conjecture.