Absolute convergence conjecture for sewn supergeometric tubes

Let $Q_1\in SK(m)$ and $Q_2\in SK(n)$ be supergeometric data of the forms

$$Q_1=((z_1,\theta_1),\ldots,(z_{m-1},\theta_{m-1});(A^{(0)},M^{(0)}),(a^{(1)},A^{(1)},M^{(1)}),\ldots,(a^{(m)},A^{(m)},M^{(m)})),$$

and

$$Q_2=((z_1',\theta_1'),\ldots,(z_{n-1}',\theta_{n-1}');(B^{(0)},N^{(0)}),(b^{(1)},B^{(1)},N^{(1)}),\ldots,(b^{(n)},B^{(n)},N^{(n)})).$$

Here $SK(k)$ denotes the space of supergeometric configurations with $k$ tubes, and $\Gamma$ and $c$ are the quantities occurring in the sewing construction. Absolute convergence conjecture. If the $i$-th tube of $Q_1$ can be sewn with the $0$-th tube of $Q_2$, then the $t^{1/2}$-series

$$e^{\Gamma(t^{-1/2}a^{(i)},A^{(i)},M^{(i)},B^{(0)},N^{(0)})c}$$

is absolutely convergent at $t^{1/2}=1$. The conjecture is needed for the stated isomorphism theorem in the nontrivial positive-central-charge case; the theorem holds without it when the relevant Neveu–Schwarz representation has zero central charge. Its status is not resolved in the supplied source context.