Götsche's polynomiality conjecture for nodal curves on surfaces

Let $S$ be a smooth, irreducible, complex projective surface, let $4\mathscr{L}4$ be a line bundle on $S$, and write $Y=|\mathscr{L}|=\mathbb{P}(H^0(S,\mathscr{L}))$ with $N=\dim Y$. For a non-negative integer $r\leq N$, let $N_r(S,\mathscr{L})$ be the degree of the locus of $r$-nodal curves in $Y$. Write

$$(\partial,k,s,x)=(\mathscr{L}^2,\mathscr{L}\mathscr{K}_S,\mathscr{K}_S^2,c_2(S)).$$

Here $\mathscr{K}_S$ is the canonical bundle and $\mathscr{L}\mathscr{K}_S$ denotes the degree of $c_1(\mathscr{L})c_1(\mathscr{K}_S)$. Götsche's polynomiality conjecture. There exist polynomials $Z_r\in\mathbb{Q}[t,u,v,w]$ of degree $r$, for $r\geq0$, such that whenever $\mathscr{L}$ is $(5r-1)$-very ample,

$$N_r(S,\mathscr{L})=Z_r(\partial,k,s,x).$$

This predicts that, under the stated ampleness condition, the node number depends only on the four Chern numbers of the polarized surface. The source notes that this conjecture was proved by Tzeng.