Existence conjecture for special Kähler structures with prescribed cubic differentials

Let $frak b=(\beta_1,\dots,\beta_k)$ be a tuple as in Definition ref{defn:moduli}, and let $\gamma>1$ be an integer. Assume that the numerical condition eqref{Eq_NecessCondMixedSing} holds. For a compact Riemann surface $\Sigma$ of genus $\gamma$ and a tuple $frak p=(p_1,\dots,p_k)$ of pairwise distinct points, let $H(fp,fb)$ denote the relevant space of special Kähler structures and cubic differentials, and let $cM_k^ell(fp,fb)$ be the associated moduli space. Existence conjecture. There exists a compact Riemann surface $\Sigma$ of genus $\gamma$ and a $k$-tuple $frak p=(p_1,\dots,p_k)$ of pairwise distinct points in $\Sigma$ such that $H(fp,fb)$, and consequently $cM_k^ell(fp,fb)$, is non-empty. This conjecture proposes existence of the prescribed cubic differentials and special Kähler structures under the stated numerical condition; the supplied text does not indicate whether it has been resolved.