Polynomiality conjecture for the leading coefficients of node polynomials

Let $C_\tau(\delta)$ denote the coefficient vector associated with a type $\tau$ in the computation of the node polynomial $N_\delta(d)$. Leading-coefficient polynomiality conjecture. Every entry of $C_\tau(\delta)$ is of the form

$$\frac{3^\delta}{\delta!}P(\delta),$$

where $P(\delta)$ is a polynomial in $\delta$. This conjectural structure is reported as verified for all types with $\operatorname{def}(\tau)\le8$, so the supplied text does not establish the assertion for all types.