Positivity conjecture for refined universal curve-counting invariants

Let $S$ be a smooth projective surface and let $L$ be a $\delta$-very ample line bundle. Let $\widetilde N^{(S,L),\delta}(y)$ be the refined universal invariant. Positivity conjecture. The Laurent polynomial $\widetilde N^{(S,L),\delta}(y)$ has non-negative coefficients:

$$\widetilde N^{(S,L),\delta}(y)\in\mathbb Z_{\ge0}[y^{\pm1}].$$

The conjecture is proved in the cited work for abelian and K3 surfaces, and the paper records further evidence from toric computations and numerical checks.