The modularity conjecture in the split Lagrangian case

Let $\mathcal{E}$ be a rank $m$ vector bundle on $X'$, let $\mathcal{E}'=\sigma^{*}\underline{\operatorname{Hom}}(\mathcal{E},\nu^{*}\mathfrak{L})$, and let $\mathcal{G}=\mathcal{E}\oplus\mathcal{E}'$ be the associated Hermitian vector bundle. Thus $(\mathcal{G},\mathcal{E})$ and $(\mathcal{G},\mathcal{E}')$ are points of $\operatorname{Bun}_{P_m,\omega_X^{-1}\otimes\mathfrak{L}}(k)$. Split Lagrangian modularity conjecture. In $\operatorname{Ch}_{r(n-m)}(\operatorname{Sht}^{r}_{U(n),\mathfrak{L}})$, one has

$$\chi(\det\mathcal{E})q^{n\deg\mathcal{E}/2}\sum_{a\in\mathcal{A}_{\mathcal{E}}(k)}\zeta_{*}[\mathcal{Z}^{r}_{\mathcal{E}}(a)] =\chi(\det\mathcal{E}')q^{n\deg\mathcal{E}'/2}\sum_{a'\in\mathcal{A}_{\mathcal{E}'}(k)}\zeta_{*}[\mathcal{Z}^{r}_{\mathcal{E}'}(a')],$$

and equivalently

$$\eta(\mathfrak{L})^{mn}q^{n(\deg\mathcal{E}-m\deg\mathfrak{L})}\sum_{a\in\mathcal{A}_{\mathcal{E}}(k)}\zeta_{*}[\mathcal{Z}^{r}_{\mathcal{E}}(a)] =\sum_{a'\in\mathcal{A}_{\mathcal{E}'}(k)}\zeta_{*}[\mathcal{Z}^{r}_{\mathcal{E}'}(a')].$$

This is the specialization of the main modularity conjecture comparing the two Lagrangian sub-bundles. Its status is therefore open in general, although it has consequences and special cases arising from the broader conjecture.