The multiplicative comparison conjecture for the mod $p$ derived Hecke algebra

Let $G$ be a locally profinite group, $K\subset G$ a compact open subgroup, and $S$ a coefficient ring. Let $S[G/K]\longrightarrow\mathbf I$ be an injective resolution, and let $\mathcal H_S(G,K)$ denote the algebra of $G$-equivariant cohomology classes defined in section. The common double-coset description identifies both constructions as $S$-modules:

$$\mathcal H_S(G,K)\longrightarrow\bigoplus_{g\in K\backslash G/K}H^n(K\cap gKg^{-1},\mathbf 1),\qquad F\longmapsto\left(F(K,gK)\right)_{g\in K\backslash G/K}.$$

Multiplicative comparison conjecture. The cohomology algebra

$$\mathbb H^*\left(\operatorname{Hom}_{S[G]}(\mathbf I,\mathbf I)^{op}\right)$$

and $\mathcal H_S(G,K)$ are isomorphic as graded algebras via this double-coset identification; equivalently, the displayed isomorphism of $S$-modules is an algebra isomorphism, with the right-hand side canonically identified with $\mathbb H^*\left(\operatorname{Hom}_{S[G]}(\mathbf I,\mathbf I)^{op}\right)$. The conjecture asserts that the two multiplication operations agree. The comparison would establish that the explicit algebra of $G$-equivariant cohomology classes agrees multiplicatively with the derived endomorphism cohomology algebra, not merely as graded $S$-modules.