Recently Andrews, Dixit, and Yee introduced smallest parts partition functions associated to the classical third order mock theta functions. One of these mock theta functions is $\omega(q)$. Shortly afterward Andrews, Dixit, Schultz, and Yee considered the overpartition analog of the smallest parts partition function associated with $\omega(q)$. In their work, they left as a conjecture the (mock) modularity of the underlying partition function associated to this overpartition smallest parts function.
In joint work with Kathrin Bringmann and Karl Mahlburg, we use a blend of techniques from q-series and harmonic weak Maass forms to address this question. In particular the underlying partition function falls short of the definition of a mock modular form, but is close.