2 problems

Espigulé–Juher–Saldaña's conjecture. Analogous holes exist in $\mathcal M_n$ for every $n>2$.

Connectedness-locus embedding conjecture. There exists a continuous map from $\mathcal{M}_{\exp}$ to $\mathcal{M}_{\mathrm{err}}$ that is a homeomorphism onto its image.