F-isomorphism conjecture for sub-Hopf algebra invariants of the Steenrod algebra

Let $\mathcal{A}$ be the mod-$2$ Steenrod algebra, let $D$ be the relevant normal sub-Hopf algebra, and let $\mathcal{E}$ be the family of sub-Hopf algebras used in the paper. For each $n$, write $P_DH_*(BV_n)$ for the $D$-primitive elements and use $[\,\cdot\,]_{GL(n)}$ for $GL(n)$-coinvariants; the superscript $\mathcal{A}/D$ denotes the corresponding quotient-Hopf-algebra invariants. The F-isomorphism conjecture. The canonical algebra homomorphisms

$$i_D^{\mathcal{A}}\colon \bigoplus_n P_{\mathcal{A}}H_*(BV_n)_{GL(n)}\longrightarrow \bigoplus_n[P_DH_*(BV_n)_{GL(n)}]^{\mathcal{A}/D},$$

and

$$i_{\mathcal{E}}^D\colon \bigoplus_n[P_DH_*(BV_n)]_{GL(n)}\longrightarrow \bigoplus_n\varprojlim_{E\in\mathcal{E}}[P_EH_*(BV_n)]_{GL(n)}$$

are both $F$-isomorphisms. These maps are isomorphisms in degree $n=1$, and the conjecture is motivated by known transfer calculations in ranks at most three and by the expected nilpotence of their kernels.