Very presentable targets have locally geometric mapping stacks

Let $T$ be a geometric $n$-stack that is very presentable in the sense that its fundamental groups over artinian bases are affine and its higher homotopy groups are vector sheaves. Let $X\to S$ be a smooth, or possibly merely flat, projective morphism. The very-presentable mapping-stack conjecture. The mapping stack $Hom(X/S,T)$ is locally geometric. This proposes a general extension of the preceding representability results to arbitrary, not necessarily connected, geometric $n$-stacks; the source gives no resolution.