The cohomological spectrum conjecture for infinitesimal unipotent supergroup schemes

Let $k$ be algebraically closed, and let $G$ be an infinitesimal unipotent $k$-supergroup scheme of height $\leq r$. Let $H(G,k)$ denote the cohomology algebra, let $\mathcal{N}_r(G)$ denote the scheme of height-$r$ one-parameter subgroups of $G$, and let

$$\psi_r:H(G,k)\longrightarrow k[\mathcal{N}_r(G)]$$

be the associated algebra homomorphism. Write $|G|=\operatorname{Max}(H(G,k))$ for the cohomological spectrum, and let

$$\Psi=\Psi_r:\mathcal{N}_r(G)\longrightarrow |G|$$

be the associated morphism. The cohomological spectrum conjecture. The kernel of $\psi_r$ is nilpotent, its image contains the $p^r$-th power of every element of $k[\mathcal{N}_r(G)]$, and consequently $\Psi$ is a homeomorphism. This conjecture is presented as an expected application of the announced BIKP detection theorem; the supplied text gives a justification showing how that theorem would imply nilpotence of the kernel, while the full statement is not reported as resolved.