Comparison conjecture for categorical Bockstein sequences

Let $k$ be a complete discrete valuation ring with uniformizing element $l\in k$. Let $c\colon G\longrightarrow k^*$ be a continuous multiplicative character of a profinite group $G$, and let $H\subset G$ be a closed normal subgroup annihilated by the reduced character $c/l\colon G\longrightarrow (k/l)^*$. For objects $X,Y\in\mathcal E_{k,0}^{G/H,+}$, suppose that the natural maps

$$\operatorname{Ext}_{{\mathcal A}_{k/l^{r+1}}^{G.+}}^n({}_{l^{r+1}}X,{}_{l^{r+1}}Y(n))\longrightarrow \operatorname{Ext}_{{\mathcal A}_{k/l^r}^{G.+}}^n({}_{l^r}X,{}_{l^r}Y(n))$$

are surjective for all $r>0$. Comparison conjecture. If the exact category ${\mathcal F}_{k/l}^{G,+}$ satisfies the main hypothesis, then the comparison functors

$$\varkappa\colon {\mathcal F}_k^G{}^+/l^r\longrightarrow {\mathcal F}_{k/l^r}^{G,+}$$

are equivalences of exact categories for all $r>0$. This predicts compatibility between reduction modulo powers of the uniformizer and the corresponding exact categories, under the stated Ext-surjectivity and main-hypothesis assumptions.