Let $X$ be a proper Deligne-Mumford stack 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 point of $X$, 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.

EDIT 1: The previous version is a bit misleading: the assumption for the statement is that for any $x\in X^T$, the local ring $\mathcal{O}_{x,X}$ is regular (but not $\mathcal{O}_{x,X^T}$).

EDIT 2: Following Ariyan's comment, I present my attempted proof for the case of schemes: Let $X'$ be the singular locus of $X$. Then $T$ acts on $X'$ and leaves each of its irreducible components invariant. For each component, we apply Borel fixed-point theorem to get a fixed point $x$. By our assumption, it is a smooth point of $X$, and hence $x\not\in X'$, a contradiction.