The extension-composition conjecture for additive homogeneous superfunctors

Let $A$ and $B$ be additive homogeneous superfunctors of degree $p^s$, with $s>0$, and finite-dimensional values. Let $F$ and $G$ be homogeneous strict polynomial functors of degree $p^r$. Here $\boldsymbol{\boldsymbol{\mathcal{P}}}$ denotes the category of strict polynomial superfunctors, and $\boldsymbol{\mathcal{P}}$ the corresponding category used for $F$ and $G$. The notation $F_{\mathrm{Ext}^*_{\boldsymbol{\mathcal{P}}}(A,B)}$ denotes parametrisation of $F$ by the graded vector space of extensions.

Extension-composition conjecture. There is a graded isomorphism, taking total degree on the left-hand side,

$$\mathrm{Ext}^*_{\mathcal{P}}\bigl(G,F_{\mathrm{Ext}^*_{\boldsymbol{\mathcal{P}}}(A,B)}\bigr)\simeq \mathrm{Ext}^*_{\mathcal{P}}(G\circ A,F\circ B),$$

natural in $A$, $B$, $F$, and $G$.

The conjecture extends the preceding result, which establishes the corresponding isomorphism when $G=I^{(r)}$. It proposes a general compatibility between extension groups of strict polynomial superfunctors and composition with additive homogeneous superfunctors; the source provides no resolution status beyond presenting this statement as a conjecture.