Let $M_{3,1}$ be the (coarse, non-compactified) moduli space of genus $3$ curves with a marked point over a field $k$ of characteristic zero. Throwing away the hyperelliptic curves, take the open subset $M_{3,1}^{nh}$ of non-hyperelliptic curves, which under the canonical embedding correspond to smooth plane quartics. The variety $M_{3,1}^{nh}$ has a stratification: $$M_{3,1}^{nh}= M_{3,1}^{generic} \sqcup M_{3,1}^{bitangent} \sqcup M_{3,1}^{flex}\sqcup M_{3,1}^{hyperflex} .$$ Here the superscript denotes the behaviour of the marked point $P$ under the canonical embedding. (For example, $P$ is a hyperflex if and only if $4P$ is a canonical divisor.) It is known by work of Looijenga ('Cohomology of M_3 and M_{3,1}') that each stratum is unirational.

Let $J \rightarrow M_{3,1}^{nh}$ be the universal Jacobian over $M_{3,1}^{nh}$. For $D$ in $$\{generic, bitangent, flex, hyperflex\},$$ write $J^D \rightarrow M_{3,1}^D$ for the pullback to the corresponding stratum.

Question.Is $J^D$ a unirational variety?

I'm especially interested in the hyperflex case. Any hints or references would be appreciated.