Quantum modularity of real-origami generating functions

Prove that the generating function for real origami in the stratum \(\mathcal{H}(\mu_1-1,\mu_1-1,\ldots,\mu_k-1,\mu_k-1)\), where \(\mu\) is a partition of \(n\), is a quantum modular form.

Background

The paper establishes an explicit divisor-sum formula for genus-two real origami with two simple zeros and observes that the generating function involving $\sigma_2(n)$ belongs to the class of quantum modular forms. This motivates extending quantum modularity from that special stratum to real-origami generating functions associated with paired zero orders. The conjecture concerns the full generating function for each stratum specified by a partition $\mu$, rather than only the genus-two example treated explicitly.