Automorphism-group conjecture for rectangular-combination designs

Let $\widetilde{\mathscr{D}}$ be a design and let ${\mathscr{D}}(\widetilde{\mathscr{D}},(e_1,e_2))$ denote the design obtained from it by the rectangular combination construction, with positive integers $e_1,e_2$. Automorphism-group conjecture. If $e_1\geq 3$, then

$$\operatorname{Aut}({\mathscr{D}}(\widetilde{\mathscr{D}},(e_1,e_2)))=\operatorname{Aut}(\widetilde{\mathscr{D}})\wr(S_{e_1}\times S_{e_2}).$$

This conjecture proposes that the exceptional enlargement of the automorphism group observed when $e_1=2$ does not occur for $e_1\geq 3$; the claim remains open in the supplied text.