Polynomial-generator conjecture for the cohomology of $K_{1+k(\sigma-1)}$

Let $K_{1+k(\sigma-1)}$ be a $C_2$-equivariant Eilenberg–Mac Lane space, with $k\geq 0$. The known initial cases are

$$H^{\ast}(K_1)=P(x_1),\qquad H^{\ast}(K_{\sigma})=P(x_{\sigma},x_{1+\sigma})/(x_{\sigma}^{2}+ax_{\sigma}+ux_{1+\sigma}).$$

Polynomial-generator conjecture. $H^{\star}(K_{1+k(\sigma-1)})$ is a polynomial algebra on $k+1$ generators with $k$ relations; the square of each of the first $k$ generators is a linear combination of the other generators. The dimensions of the first $k$ generators are obtained recursively by adding $\sigma-1$ to the dimensions of the generators for H^{\star}(K_{1+(k-1)(\sigma-1)})}, and the last generator has dimension

$$k\sigma+2^k-k.$$

This conjecture would provide the inductive input for applying the Borel theorem to compute $H^{\star}(K_{1+m+k(\sigma-1)})$ for $m\geq 0$. The stated recursive pattern is illustrated by the case $k=3$, but the source gives no resolution of the conjecture.