The support-variety detection conjecture for height-1 infinitesimal unipotent supergroups

Suppose $k$ is algebraically closed. Let $G$ be a height-$1$ infinitesimal unipotent $k$-supergroup scheme, let $M$ be a finite-dimensional rational $G$-supermodule, and let $\mathcal{N}_1(G)_M$ denote the subset of $\mathcal{N}_1(G)$ on which the pullback $\phi^*M$ has infinite projectivity measure for the odd one-parameter supergroup $\mathbb{M}_1$. Let $|G|_M$ denote the support variety of $M$, and let

$$\Psi=\Psi_1:\mathcal{N}_1(G)\longrightarrow |G|$$

be the homeomorphism asserted by the cohomological spectrum conjecture. The support-variety detection conjecture. Under this homeomorphism,

$$\Psi^{-1}(|G|_M)=\mathcal{N}_1(G)_M=\left\{\phi\in\mathcal{N}_1(G):\operatorname{id}_{\mathbb{M}_1}(\phi^*M)=\infty\right\}.$$

This is presented as a consequence expected from the BIKP detection theorem and the cohomological spectrum conjecture; the supplied text does not report a resolution.