The BPS cohomological interpretation of the p-adic integral for $SL_n$ Higgs bundles

Let $S=\operatorname{Spec}\mathcal{O}_F$ where $\mathcal{O}_F$ is a local ring with finite residue field $k_F$. Let $D=K_\mathcal{C}$ and let $L\in\mathcal{P}ic^d(\mathcal{C})$ for $d\in\mathbb{Z}$. Let $\alpha$ denote the gerbe

$$\mathcal{M}^{L}_{\widetilde{SL_n}}\longrightarrow\mathcal{M}^{L,\operatorname{rig}}_{SL_n}.$$

Set $q=|k_F|$. The BPS cohomological interpretation conjecture. Using the notation for Frobenius introduced in the paper,

$$\int_{M_{SL_n}^{L}(\mathcal{O}_F)^\sharp} f_\alpha\,\mu_{\operatorname{can}}=q^{-\dim M_{SL_n}^{L}}\operatorname{tr}\left(\operatorname{Fr}\mid H^*_{BPS}(\mathcal{L}M_{SL_n,k_F}^{L})_1\right).$$

This conjecture proposes a BPS-cohomological interpretation of the $p$-adic integral in the canonical-bundle case. It is motivated by the Hausel--Thaddeus-type BPS conjecture, but remains speculative because the BPS sheaf has not yet been constructed as an $\ell$-adic sheaf.