Modularity conjecture for higher theta series of unitary shtukas

Let $m\le n$ be fixed. Let $X'/X$ be the given double cover, let $\operatorname{U}_{2m}$ be the associated quasi-split unitary group, and let $\operatorname{Bun}_{\operatorname{U}_{2m}}(k)$ denote its $k$-points. The function $\widetilde{Z}^{r}_{m}$ is defined on the parabolic moduli and takes values in $\operatorname{Ch}_{r(n-m)}(\operatorname{Sht}^{r}_{\operatorname{U}_{n}})$. Modularity conjecture. The function $\widetilde{Z}^{r}_{m}$ descends to a function

$$Z^{r}_{m}:\operatorname{Bun}_{\operatorname{U}_{2m}}(k)\to \operatorname{Ch}_{r(n-m)}(\operatorname{Sht}^{r}_{\operatorname{U}_{n}}).$$

This asserts that the higher theta series constructed from the special cycles is invariant under the passage from the Siegel parabolic moduli to the moduli of the full unitary group, providing a modularity property for these algebraic cycle-valued generating series.