Categorical trace conjecture for the diagonal correspondence

Let $H$ be a reductive group, let $G=H\times H$, and let $X=H\backslash(H\times H)$. Let $c:H\to H$ be the Cartan involution and let $fs_H$ be a local system on the dual group. Set $fs=(fs_H,c^*fs_H)$ and choose a Hecke eigensheaf $\LL_{fs_H}$ on $\Bun_H$. Its Verdier dual $\DD(\LL_{fs_H})$ is a Hecke eigensheaf with eigenvalue $c^*fs_H$, and set $\LL_{fs}=\LL_{fs_H}\boxtimes\DD(\LL_{fs_H})$. Let $e\in\pi_1(H)$, let $V_H\in\operatorname{Rep}(\Hc^I)$ be irreducible, and put $V^I=V_H^I\boxtimes V_H^I$. Let $\xi_{fs,I}^e:V_{fs}^I\to l_{H,I,!}(\operatorname{IC}_{V_H^I}|_{\Sht_{H,I}^e})\otimes l_{H,I,!}(\operatorname{IC}_{V_H^I}|_{\Sht_{H,I}^e})$ be the isotypic-part map, and let $\frc$ be the diagonal cohomological correspondence induced by the Hecke stack. Write $\ev_{V_{H,fs_H}^I}:V_{fs}^I\cong V_{H,fs_H}^I\otimes V_{H,fs_H}^{I*}\to\uk_{C^I}$ for evaluation. Categorical trace conjecture.

$$\tr_{\Sht,C^I}(\frc)\circ\xi_{fs,I}^e=\tr(\Frob,\Gamma_c(\LL_{fs_H}^e\otimes\DD(\LL_{fs_H}^e)))\cdot\ev_{V_{H,fs_H}^I}.$$

This predicts that the trace of the diagonal correspondence on the Shtuka isotypic part is the Frobenius trace on the compactly supported cohomology of the eigensheaf tensor its Verdier dual, multiplied by the natural representation-theoretic evaluation map. The supplied context gives no resolution status.