The boundary-value conjecture for LCS structures and Green functions

Let $S$ be a compact complex surface in class $VII_0$, and let $\mathcal T(S)=]a,b[$, with $-\infty\leq a$ and $b<0$, be the moduli space of locally conformally symplectic structures taming the complex structure of $S$. Let $\varphi$ be the decreasing map

$$\varphi:]-\infty,0]\to [1,+\infty[,$$

with $\varphi(c)=e^{-c}$. A boundary-value conjecture asserts that if there is a Green function $\hat u$ on the cyclic covering $\hat S$ with multiplicative constant $\lambda>1$, or a twisted logarithmic $1$-form $\theta\in H^0(S,\Omega^1(\log D)\otimes\mathcal L_\lambda)$ with twisting constant $\lambda>1$, then

$$\lambda=\varphi(b).$$

This gives a geometric interpretation of the upper endpoint of the moduli interval of LCS structures; the supplied text presents it as a conjectural meaning of that endpoint, and does not state a resolution.