Dualizability conjecture for Hecke-lisse sheaves with level structure

With $X$, $G$, $\ul{x}$, and $\ul{\operatorname{Shv}}_{\operatorname{HL}}(\operatorname{Bun}_G^{\operatorname{level}_{\ul{x}}})$ as above, regard this object as an object of $\fL(G)_{\ul{x}}\operatorname{Cat}$. Dualizability conjecture. The object $\ul{\operatorname{Shv}}_{\operatorname{HL}}(\operatorname{Bun}_G^{\operatorname{level}_{\ul{x}}})$ is dualizable in $\fL(G)_{\ul{x}}\operatorname{Cat}$. This technical conjecture is used to define its Frobenius trace; the source gives no resolution evidence.