Trivializations of principal bundle over a $\infty$-topos

Let $G$ be a group object in an $\infty$-topos $\mathcal T$, and let $P \to X$ be a $G$-torsor as in the accepted answer by Daniël Apol to this question; equivalently, as in the answer, such a $G$-torsor is given by a map $\varphi \colon X \to BG$, and $P$ is the pullback of $\varphi$ along the canonical map $\ast \to \ast/G = BG$.

My question is: do we then know if, locally in the topology on $\mathcal T$, $P$ has the form $X \times G$? i.e., does there exist a cover $\{U_i\}$ of $X$, such that the pullback of $P$ to $U_i$ is the trivial principal bundle $U_i \times G$?

This particularly interests me in the case of $\mathcal T$ being the topos of étale $\infty$-stacks, and $G = B\mathbb G_m$ being the classifying space of the multiplicative group.