Conjecture II on Ext-dimension comparison for algebraic and quantum groups

Let $G$ be a semisimple algebraic group with root system as in the preceding conjecture, let $p$ be KL-good, and let $\Gamma$ be a finite poset ideal of dominant weights. Set $A:=\operatorname{Dist}(G)_\Gamma$, a finite-dimensional quasi-hereditary algebra. Let $\ell(p)=p$ for odd $p$ and $\ell(p)=4$ for $p=2$, let $\zeta$ be a primitive $\ell(p)$th root of unity, and let $B=U_{\zeta,\Gamma}$ be the corresponding finite-dimensional algebra for the quantum enveloping algebra $U_\zeta$. Denote the standard, costandard and irreducible modules by $\Delta(\lambda)$, $\nabla(\lambda)$, $L(\lambda)$ for $A$, by $\Delta^{\operatorname{red}}(\lambda)$ and $\nabla_{\operatorname{red}}(\lambda)$ for the relevant reduced modules, and by $\Delta_\zeta(\lambda)$, $\nabla_\zeta(\lambda)$ and $L_\zeta(\lambda)$ for $B$.

Conjecture II. For all $\lambda,\mu\in\Gamma$ and all $n\geq 0$, the following equalities hold:

$$\begin{cases} \dim\operatorname{Ext}^n_B(\Delta_\zeta(\lambda),L_\zeta(\mu))=\dim\operatorname{Ext}^n_A(\Delta(\lambda),\nabla_{\operatorname{red}}(\mu)),\\ \dim\operatorname{Ext}^n_B(L_\zeta(\lambda),\nabla_\zeta(\mu))=\dim\operatorname{Ext}^n_A(\Delta_{\operatorname{red}}(\lambda),\nabla(\mu)),\\ \dim\operatorname{Ext}^n_B(L_\zeta(\lambda),L_\zeta(\mu))=\dim\operatorname{Ext}^n_A(\Delta_{\operatorname{red}}(\lambda),\nabla_{\operatorname{red}}(\mu)). \end{cases}$$

The conjecture asserts that Ext-group dimensions for the algebraic group and the quantum group at the specified root of unity agree after replacing modules on the algebraic side by the corresponding reduced modules. The paper explains that, when Conjecture I holds, this comparison reduces to further graded Ext statements.