Let $\mathcal{X}$ be an algebraic stack, and $x \in |\mathcal{X}|$ a point such that the residual gerbe $\mathcal{G}_x$ of $\mathcal{X}$ at $x$ exists. Let $\mathcal{Y}$ be a reduced algebraic stack, and $\mathcal{Y} \to \mathcal{X}$ a morphism such that the induced map on topological spaces factors through the inclusion $x \to |\mathcal{X}|$. Does the map $\mathcal{Y} \to \mathcal{X}$ factor through $\mathcal{G}_x \to \mathcal{X}$? If not, are there any additional conditions one can place on $\mathcal{Y}, \mathcal{X}$, or the map $\mathcal{Y} \to \mathcal{X}$ to get an affirmative answer?

I require $\mathcal{Y}$ to be reduced above as there are easy counterexamples to the above when $\mathcal{Y}$ is nonreduced (e.g. the identity map on the dual numbers).