  The Chow ring of $\mathcal{M}_g$ is known to be generated by tautological
classes for $g \leq 9$. Meanwhile, the first example of a non-tautological
class on $\mathcal{M}_{g}$ is the fundamental class of the bielliptic locus in
$\mathcal{M}_{12}$, due to van Zelm. It remains open if the Chow rings of
$\mathcal{M}_{10}$ and $\mathcal{M}_{11}$ are generated by tautological
classes. In these cases, a natural first place to look is at the bielliptic
locus. In genus $10$, it is already known that classes supported on the
bielliptic locus are tautological. Here, we prove that all classes supported on
the bielliptic locus are tautological in genus $11$. By Looijenga's vanishing
theorem, this implies that they all vanish.
