Flat projective mapping stacks are slightly geometric

Let $T$ be any geometric, locally geometric, or slightly geometric $n$-stack, and let $X\to S$ be a flat projective morphism. The slightly geometric mapping-stack conjecture. The mapping stack $Hom(X/S,T)$ is slightly geometric. This is offered as a broader, “slightly geometric” replacement after the source notes that analogous assertions for $K({\bf G}_m,i)$ with $i\geq 3$ cannot yield locally finite-type mapping stacks; the source gives no resolution.