Higher-vector-homotopy mapping stacks are locally geometric

Let $T$ be a geometric $n$-stack whose homotopy sheaves $\pi_i$ are vector sheaves for every $i\geq 3$, and let $X\to S$ be the morphism under consideration. The higher-vector-homotopy conjecture. The mapping stack $Hom(X/S,T)$ is locally geometric. The statement is proposed by combining the preceding conjectures; no resolution is given in the source.