Let $X$ be an stable genus four curve obtained by gluing a hyprelliptic genus three curve at two points which are not hyperelliptic involution of each other. This curve corresponds to a point on the moduli space $\overline{M_4}$ which lies in the closure of the Petri locus. Assume $X$ has no automorphisms. Then is the closure of the Petri locus at this point smooth? (I would guess YES, by looking at the Hurwitz schemes that covers the $\overline{M_4}$, but I don't know how to prove it.)
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