Extension-class conjecture for general linear supergroup schemes

Assume that $k$ is a perfect field of characteristic $p>2$, let $m,n,r\in\mathbb{N}$, and write $GL(m|n)_1$ for the first Frobenius kernel of the general linear supergroup scheme. Let $\mathfrak{gl}(m|n)_{\overline{0}}$ and $\mathfrak{gl}(m|n)_{\overline{1}}$ denote its even and odd parts, respectively, and let $(\mathord{-})^{(r)}$ denote the $r$-th Frobenius twist. Cohomology classes in $\operatorname{H}^i(G,M)$ induce graded-superalgebra homomorphisms from the symmetric algebra on the appropriately shifted Frobenius twist of $M^*$ to $\operatorname{H}^\bullet(G_1,k)$. Extension-class conjecture. There exist classes

$$e_r^{m,n}\in\operatorname{H}^{2p^{r-1}}\bigl(GL(m|n),(\mathfrak{gl}(m|n)_{\overline{0}}^*)^{(r)}\bigr),\qquad c_r^{m,n}\in\operatorname{H}^{p^r}\bigl(GL(m|n),(\mathfrak{gl}(m|n)_{\overline{1}}^*)^{(r)}\bigr)$$

that restrict nontrivially to $GL(m|n)_1$. If $m,n\not\equiv0\pmod p$, the homomorphism induced by $e_r^{m,n}$ is the composition of the $p^{r-1}$-power map

$$S\bigl(\mathfrak{gl}(m|n)_{\overline{0}}^*(2p^{r-1})\bigr)^{(r)}\longrightarrow S\bigl(\mathfrak{gl}(m|n)_{\overline{0}}^*(2)\bigr)^{(1)}$$

with the horizontal edge map of the May spectral sequence. Under the same condition, composing the homomorphism induced by $c_r^{m,n}$ with the vertical edge map has image equal to the subalgebra of $\Lambda_s(\mathfrak{gl}(m|n)^*)$ generated by all $p^r$-th powers of elements of the subalgebra $S(\mathfrak{gl}(m|n)_{\overline{1}}^*)$. The classes are intended to provide the extension classes needed to establish cohomological finite generation for infinitesimal supergroup schemes; their existence and the stated restriction properties are presented as conjectural here.