Refined invariant generating-function conjecture for weighted projective planes

Let $N_\delta(d,m;y)$ be the polynomial from the weighted-projective refined node polynomial conjecture, let $\widetilde N^{(\Sigma_m,dH),\delta}(y)$ be the refined invariant on the minimal resolution $\Sigma_m$, and let $\widetilde{DG}_2$ be the refined modular series used as the change-of-variables parameter. Weighted-projective refined generating-function conjecture. There exist power series $C_1,C_2,C_3\in\mathbb Q[y^{\pm1}][[q]]$ such that

$$\sum_{\delta\ge0}N_\delta(d,m;y)(\widetilde{DG}_2)^\delta= \left(\sum_{\delta\ge0}\widetilde N^{(\Sigma_m,dH),\delta}(y)(\widetilde{DG}_2)^\delta\right)C_1^{(m+2)d}C_2^{m+2}C_3.$$

This conjecture proposes a universal multiplicative correction relating the weighted-projective and resolution-side generating functions.