Comparison conjecture for perverse Eilenberg–MacLane spaces

Let ${\mathtt P}=[n]$, and let $\overline{p}$ and $\overline{q}$ be Goresky–MacPherson perversities. Let ${\mathtt K}(R,n,{\mathtt P},\overline{p})$ denote the perverse Eilenberg–MacLane space, and let $\mathscr H^m_{\overline{q}}$ denote perverse cohomology. The natural chain injection from ordinary cochains into perverse cochains induces the comparison map below. Comparison conjecture. All perverse cohomological operations come from the classical cohomology situation; equivalently, the induced map

$$\mathscr H^m_{\overline{q}}({\mathtt K}(R,n,{\mathtt P},\overline{p});R)\longrightarrow H^m(K(R,n);R)$$

is injective. For ${\mathtt P}=[1]$, the paper proves this conjecture in degrees $m\leq n$, for arbitrary $R$, $\overline{p}$, and $\overline{q}$; the general case remains open.