Convergence conjecture for pseudo-$q$-traces of products of intertwining operators

Let $V$ be a vertex operator algebra satisfying the conditions in the earlier conjecture on products of twisted intertwining operators, and let $G$ be a finite group of automorphisms of $V$. For $g_{1},g_{2},g_{3},g_{4}\in G$, consider the pseudo-$q$-trace series of products of intertwining operators, with $q=q_{\tau}=e^{2\pi i\tau}$. Convergence conjecture. The series is absolutely convergent in the region

$$1>|q_{z_{1}}|>|q_{z_{2}}|>|q_{\tau}|>0,$$

and analytically extends to a multivalued analytic function in the region

$$\operatorname{Im}(\tau)>0,\qquad z_{1}\ne z_{2}+k\tau+l\quad (k,l\in\mathbb{Z}).$$

Moreover, each singular point $z_{1}=z_{2}+k\tau+l$ is regular: every branch of the multivalued analytic function has, in a neighborhood of that point, an expansion of the form

$$\sum_{p=0}^{K}\sum_{j=1}^{M}(z_{1}-z_{2}+k\tau+l)^{r_{j}}\bigl(\log(z_{1}-z_{2}+k\tau+l)\bigr)^{p}f_{j,p}(z_{1}-z_{2}+k\tau+l),$$

where $r_{j}\in\mathbb{R}$ for $j=1,\ldots,M$, and each $f_{j,p}(z)$, for $j=1,\ldots,M$ and $p=0,\ldots,K$, is analytic on a disk containing $0$. This conjecture concerns the analytic behavior of pseudo-$q$-traces in orbifold conformal field theories; the source attributes it to prior work on open problems and orbifold theory, and the status is not resolved in the supplied text.