Coefficient polynomiality threshold conjecture for refined Severi degrees

Fix $\delta\ge1$, and write the refined plane Severi degree as

$$N^{d,\delta}(y)=p^\delta_{d,0}y^\delta+p^\delta_{d,1}y^{\delta-1}+\cdots+p^\delta_{d,\delta}y^0+\cdots+p^\delta_{d,0}y^{-\delta}.$$

Let $N_\delta(d;y)$ be the refined node polynomial, with coefficients $p^\delta_i(d)$ defined analogously. Coefficient polynomiality threshold conjecture. For $0\le i\le\delta$, one has

$$p^\delta_i(d)=p^\delta_{d,i}$$

whenever $d\ge i+2$. Thus the $i$th coefficient stabilizes at a degree bound independent of the total number of nodes $\delta$. The paper notes that this is part of a conjecture of Göttsche–Shende.