Automorphism refinement conjecture for supercharacter theories of $C_p\times C_p$

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. Let $H$ be a nontrivial $\mathsf{S}$-normal subgroup and write $\mathsf{S}_H=[H]_m$, where $[H]_m$ is the partition into orbits of the $m$th-power automorphism. Let $\mathsf{T}$ be the supercharacter theory of $G$ coming from the automorphism sending an element to its $m$th power.

Automorphism refinement conjecture. Then

$$\mathsf{T}\preccurlyeq\mathsf{S}.$$

This predicts that every such supercharacter theory contains the automorphism-generated theory determined by its restriction to a normal subgroup. It would substantially reduce the possible structures of supercharacter theories in this setting; it is presented as open despite verification in small and computationally accessible cases.