The polarization-independence conjecture for refined Vafa–Witten generating functions

Let $S$ be the surface under consideration, let $c_1(S)$ denote its first Chern class, and let $v_0(S)$ be the null vector specified in the source. Write $\operatorname{Span}(c_1(S),v_0(S))^+$ for the part of their span lying in the Kähler cone, and let $J$ be a polarization in this cone. Let $h^S_{N,\mu,J}$ denote the generating function and let $\Theta^S_{N,\mu}(\tau,z;\{\Phi_n(J)\})$ be the corresponding theta-series expression with kernels evaluated at $J$. Polarization-independence conjecture. For every $J\in\operatorname{Span}(c_1(S),v_0(S))^+$, one has

$$h^S_{N,\mu,J}=\Theta^S_{N,\mu}(\tau,z;\{\Phi_n(J)\}).$$

This extends the canonical-chamber formula to a more general polarization and predicts that the refined Vafa–Witten generating functions retain the same theta-series structure throughout the specified part of the Kähler cone. The supplied text gives no resolution status for this assertion.