Automorphism conjecture for supercharacter theories with a non-coarsest restriction

Let $G=C_p\times C_p$, and let $\mathsf{S}$ be a supercharacter theory of $G$. Suppose that $G$ has at least three nontrivial, proper $\mathsf{S}$-normal subgroups, and let $H$ be one of them. Write $\mathsf{M}(H)$ for the coarsest supercharacter theory of $H$.

Automorphism conjecture. If $\mathsf{S}_H\neq\mathsf{M}(H)$, then every subgroup of $G$ is $\mathsf{S}$-normal. In particular, $\mathsf{S}$ comes from automorphisms.

This would show that a non-coarsest restriction to one normal subgroup forces a highly structured supercharacter theory; the paper reports verification of the related conjecture for primes up to $47$, but the general claim remains open.