Let $X$ be a stable hyperelliptic genus four curve (meaning on the moduli space it lies in the closure of the hyperelliptic locus) and $L$ a line bundle on $X$. Suppose that the pair $(X,L)$ comes from the degeneration of a $\mathfrak{g}_3^1$ on a smooth curve. This means the following. There is a DVR $R$ and a stable curve $\mathcal{X} \to Spec R$ and a line bundle $\mathcal{L}$ on $\mathcal{X}$. The special fiber of $\mathcal{X}$ is $X$ and the restriction of $\mathcal{L}$ to the special fiber is $L$. The generic fiber is a smooth genus four curve $Y$ and the restriction of $\mathcal{L}$ to $Y$ satisfies $h^0(Y, \mathcal{L}|_Y) \geq 2$.

The question is: can we take $\mathcal{X}$ to be hyperelliptic, i.e. the generic fiber $Y$ is also hyperellitpic (and smooth)?