Quantum modularity conjecture for generating functions of real origamis

Let $\mu=[\mu_1,\ldots,\mu_k]$ be a partition of $n$. The stratum $\mathcal{H}(\mu_1-1,\mu_1-1,\ldots,\mu_k-1,\mu_k-1)$ parametrizes Abelian differentials with the indicated zeros, and consider the generating function counting real origamis in this stratum.

Quantum modularity conjecture. This generating function is a quantum modular form.

The conjecture is motivated by explicit computations in low-genus strata and by the analogous quasimodularity result for complex origamis. Its general validity is not established in the supplied text.