Maximal gonality on strata of differentials and uniruledness of strata in low genus

We prove that for a generic element in a nonhyperelliptic component of an abelian stratum ${\mathcal{H}}_g(\mu )$ in genus $g$, the underlying curve has maximal gonality. We extend this result to the case of quadratic strata when the partition $\mu$ has positive entries. As a consequence we deduce that all nonhyperelliptic components of ${\mathcal{H}}_9(\mu )$ are uniruled when $\mu$ is a positive partition of 16 and all nonhyperelliptic components of ${\mathcal{H}}_g^2(\mu )$ are uniruled when $\mu$ is a positive partition of $4g-4$ and either $3\le g\le 5$ or $g=6$ and $l(\mu )\ge 4$.