Göttsche–Shende refined Severi degree conjecture

For a projective surface $S$ and line bundle $L$, let $N^{(S,L),\delta}(y)$ denote the refined Severi degree and let $\widetilde N^{(S,L),\delta}(y)$ denote the refined universal invariant. Göttsche–Shende's refined Severi degree conjecture. If $S$ is $\mathbb P^2$ or a rational ruled surface, and $\mathbb P^\delta\subset |L|$ contains no non-reduced curves and no curves containing components with negative self-intersection, then

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

The source gives explicit bounds for $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, and $\Sigma_m$, but the displayed $\Sigma_m$ formula contains a typographical error. The conjecture is supported by computations and is proved in the paper for the stated low-node ranges.