Automorphism-group conjecture for wreath-product designs

Let $\widetilde{\mathscr{D}}$ be the input design, let $e$ and $\widetilde{k}$ be the parameters of Construction $\mathscr{D}(\widetilde{\mathscr{D}},e)$, and let $\operatorname{Aut}$ denote the automorphism group. Automorphism-group conjecture. If $e\geq 3$ or $\widetilde{k}\geq 3$, then

$$\operatorname{Aut}(\mathscr{D}(\widetilde{\mathscr{D}},e))=\operatorname{Aut}(\widetilde{\mathscr{D}})\wr S_e.$$

Computations suggest that the displayed equality holds in most small examples, with the exceptional cases described in the surrounding discussion; the conjecture concerns whether those exceptions are the only ones.