Locally geometricity conjecture for $K({\bf G}_m,2)$ and finite-type group schemes

Let $S$ be a base scheme, let $X\to S$ be a flat projective morphism, and let $K({\bf G}_m,2)$ denote the Eilenberg–Mac Lane stack. For any group scheme $G$ of finite type, let $BG$ denote its classifying stack. The $K({\bf G}_m,2)$ and finite-type group-scheme conjecture. The mapping stack $Hom(X/S,K({\bf G}_m,2))$ is locally geometric; similarly, $Hom(X/S,BG)$ is locally geometric for every group scheme $G$ of finite type, including an abelian variety. The claim is presented as a concrete extension of the preceding conjecture, and the source gives no resolution.