Smooth toric refined Severi degree conjecture

Let $\Delta$ be a convex lattice polygon, let $S=X(\Delta)$ be the associated smooth toric surface, and let $L=L(\Delta)$ be a $\delta$-very ample line bundle. Let $N^{\Delta,\delta}(y)$ be the tropical refined Severi degree and $\widetilde N^{(S,L),\delta}(y)$ the refined universal invariant. Smooth toric refined Severi degree conjecture.

$$N^{\Delta,\delta}(y)=\widetilde N^{(S,L),\delta}(y).$$

The paper proves this comparison for several basic toric surfaces and leaves the general smooth toric case conjectural.