Faithfulness of tensoring by a quasi-coherent sheaf on Deligne–Mumford stacks

Let $X$ be a quasi-separated Deligne–Mumford stack with generically trivial stabilizer. Let $\mathcal{M}\in \operatorname{QCoh}(X)$ and let $\alpha\in \mathrm{Br}(X)$. Write $\operatorname{QCoh}(X,\alpha)$ for the category of $\alpha$-twisted quasi-coherent sheaves on $X$. Tensor-action faithfulness conjecture. The functor

$$-\otimes_{\mathcal{O}_{X}}\mathcal{M}\colon \operatorname{QCoh}(X,\alpha)\longrightarrow \operatorname{QCoh}(X,\alpha)$$

is naturally isomorphic to the identity functor $\operatorname{id}$ if and only if $\mathcal{M}\cong \mathcal{O}_{X}$. The statement would extend the corresponding faithfulness result from varieties to Deligne–Mumford stacks with generically trivial stabilizer; the source notes that the preceding results fail for some Deligne–Mumford stacks and asks whether examples with generically trivial stabilizers exist.