Feng-Yun-Zhang's modularity conjecture for special cycles on unitary shtuka spaces

Assume $n=2m$. Let $F=k(X)$ and $F'=k(X')$, let $\chi\colon \mathbb A_{F'}^\times\to\mathbb C^\times$ be an unramified Hecke character trivial on $\mathbb A_F^\times$, and let $H_m$ be the rank $2m$ quasi-split unitary group over $F$ with standard maximal compact subgroup $K_m$. For a rank-$m$ vector bundle $\mathcal E$ on $X'$ and a hermitian morphism $a\colon\mathcal E\to\sigma^*\mathcal E^\vee$, set

$$d(\mathcal E)=\frac{\deg(\mathcal E^\vee)-\deg(\mathcal E)}{2}=m\deg(\omega_X)-\deg(\mathcal E).$$

Feng-Yun-Zhang's modularity conjecture. There is an automorphic form

$$Z^{r,\chi}\colon H_m(F)\backslash H_m(\mathbb A_F)/K_m\to\operatorname{Ch}^{rm}(\operatorname{Sht}^r_{U(n)})_\mathbb C$$

whose geometric Fourier coefficients are

$$Z^{r,\chi}_{(\mathcal E,a)}=\frac{\chi(\det(\mathcal E))}{q^{m d(\mathcal E)}}[\mathcal Z^r_{\mathcal E}(a)]\in\operatorname{Ch}^{rm}(\operatorname{Sht}^r_{U(n)})_\mathbb C.$$

This is the middle-codimension case of the modularity conjecture cited in the source; it predicts that the special-cycle classes assemble into an automorphic form, while the meaning of $\chi(\det(\mathcal E))$ is supplied by the Eisenstein-series construction.