Infinite free-extension naive lifting conjecture

Let $A$ be a divided power DG $R$-algebra, and let

$$B=A\langle X_i\mid i\in\mathbb{N}\rangle.$$

Let $N$ be a bounded below semifree DG $B$-module.

Infinite free-extension naive lifting conjecture. If

$$\operatorname{Ext}^{i}_{B}(N\oplus B,N\oplus B)=0$$

for all $i\geqslant 1$, then $N$ is naively liftable to $A$.

The conjecture extends the paper's finite free-extension lifting results to an infinite free extension and is posed because the authors need this case in connection with the Auslander–Reiten conjecture. The source states that no proof is known.