# Do gerbes induce surjections on etale fundamental groups?

Let $$f : \mathcal{X}\rightarrow\mathcal{Y}$$ be a morphism of algebraic stacks which is a gerbe in the sense of the stacks project. Must $$f$$ induce a surjection of etale fundamental groups? I'm happy to assume that $$\mathcal{Y}$$ admits a coarse space $$Y$$.

By Theorem 7.11 in Noohi, the coarse map $$\mathcal{Y}\rightarrow Y$$ induces a surjection on fundamental groups, so for my question it would suffice to show that the composition $$\mathcal{X}\rightarrow\mathcal{Y}\rightarrow Y$$ is also a coarse space for $$\mathcal{X}$$. Is this true? Gerbes are universal homeomorphisms, and at least if we work with the etale topology then gerbes induce bijections on sets of $$k$$-points for algebraically closed fields $$k$$, but it's not clear to me why the universal property of coarse spaces must hold.

If not, are there any reasonable positive results?