Free perverse algebra conjecture for perverse Eilenberg–MacLane spaces

Let ${\mathtt P}=[n]$, let $\overline{p}$ be a Goresky–MacPherson perversity, and let $m$ be the differential degree of a generator $x$. Write $\mathscr H^\ast_{\overline{\bullet}}$ for the perverse cohomology algebra. Free perverse algebra conjecture. The following isomorphisms of perverse algebras should hold:

$$\mathscr H^\ast_{\overline{\bullet}}(K(\mathbb{Q},m,[n],\overline{p});\mathbb{Q})\cong \bigwedge_{\overline{\bullet}} x,$$

where $\bigwedge_{\overline{\bullet}}x$ is the free rational commutative graded perverse algebra on one generator of differential degree $m$ and perverse degree $\overline{p}$, and

$$\mathscr H^\ast_{\overline{\bullet}}(K(\mathbb{F}_{2},m,[n],\overline{p});\mathbb{F}_{2})\cong \mathscr K^\ast_{\overline{\bullet}}(x),$$

where $\mathscr K^\ast_{\overline{\bullet}}(x)$ is the free unstable perverse algebra on one generator of differential degree $m$ and perverse degree $\overline{p}$. These formulas would identify the perverse cohomology algebras with the free algebras generated by the fundamental class, extending the classical rational and mod-$2$ calculations; the statement is presented conditionally in the source and remains open here.