  We describe graded commutative Gorenstein algebras ${\mathcal E}_n(p)$ over a
field of characteristic $p$, and we conjecture that
$\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}}(1,1)\cong{\mathcal E}_{n}(p)$,
where $\mathsf{Ver}_{p^{n+1}}$ are the new symmetric tensor categories recently
constructed in \cite{Benson/Etingof:2019a,Benson/Etingof/Ostrik,Coulembier}. We
investigate the combinatorics of these algebras, and the relationship with
Minc's partition function, as well as possible actions of the Steenrod
operations on them.
  Evidence for the conjecture includes a large number of computations for small
values of $n$. We also provide some theoretical evidence. Namely, we use a
Koszul construction to identify a homogeneous system of parameters in
${\mathcal E}_n(p)$ with a homogeneous system of parameters in
$\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}}(1,1)$. These parameters have
degrees $2^i-1$ if $p=2$ and $2(p^i-1)$ if $p$ is odd, for $1\le i \le n$. This
at least shows that $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}}(1,1)$ is a
finitely generated graded commutative algebra with the same Krull dimension as
${\mathcal E}_n(p)$. For $p=2$ we also show that
$\mathrm{Ext}^\bullet_{\mathsf{Ver}_{2^{n+1}}}(1,1)$ has the expected rank
$2^{n(n-1)/2}$ as a module over the subalgebra of parameters.
