The fiber-base duality conjecture for Vafa-Witten generating functions

Let $\Lambda_S$ be the lattice of a del Pezzo surface. Let $v_0,v'_0\in\Lambda_S$ be null vectors satisfying

$$v_0\cdot v'_0=1,\qquad r_1\equiv c_1\cdot v_0=c_1\cdot v'_0.$$

Let $\{E_I\}_{I=1}^{b_2-2}$ be an orthonormal basis of the orthogonal complement of $v_0$ and $v'_0$, and set

$$\kappa=\frac{r_1}{2},\qquad \kappa_I=|c_1\cdot E_I|.$$

The parameters satisfy the condition stated in the source. Fiber-base duality conjecture. The two sets of generating functions constructed from $v_0$ and $v'_0$, using the stated construction with $\phi_{N,\mu}$ given by the universal holomorphic ambiguity, are equal:

$$g_{N,\mu}[v_0]=g_{N,\mu}[v'_0].$$

This generalizes the fiber-base duality relation from Hirzebruch surfaces to del Pezzo surfaces and is supported in the source by explicit checks and numerical evidence, but is not proved in general.