Let $X$ be a proper Deligne-Mumford stack of finite type over $\mathbb{C}$ with an action by a complex torus $T$. Let $X^T$ denote the fixed locus.

Question: Is the following statement true?

If every point of $X^T$ is a smooth, then $X$ is smooth.

If $X$ is a scheme, I know the answer is yes and it is proved by applying the Borel fixed-point theorem to the singular locus of $X$. So the question will be solved if the fixed-point theorem holds for DM stacks. But I could not find a reference for this.