Propagation of the main hypothesis through categorical Bockstein reduction

Let $k$ be a complete discrete valuation ring with uniformizing element $l\in k$. 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 $r>0$. Propagation conjecture. The exact categories ${\mathcal F}_{k/l^r}^G$ satisfy the main hypothesis whenever the exact category ${\mathcal F}_{k/l}^G$ does. This asserts that the main homological condition propagates from reduction modulo $l$ to every reduction modulo a positive power of $l$.