The one-step quantum Frobenius Ext decomposition conjecture

Let $F$ and $G$ be strict polynomial functors, let $F^{(1)_q}$ and $G^{(1)_q}$ denote their quantum Frobenius twists, and let $E_1=\operatorname{Ext}_{\mathcal{P}^{}_q}^*(I^{(1)_q},I^{(1)_q})$. For each degree $k$, the one-step quantum Frobenius Ext decomposition conjecture. There is a graded isomorphism

$$\operatorname{Ext}^k_{\mathcal{P}^{}_q}(F^{(1)_q},G^{(1)_q})\simeq\bigoplus_{i+j=k}\operatorname{Ext}^i(F,G^j(E_1\otimes I)).$$

The paper presents the theorem on the quantum Frobenius twist of the identity and its small-characteristic analogue as special cases of this broader conjecture; a complete solution is not established here.