Generating-series conjecture for quasihomogeneous Hilbert schemes of points

Let $a,b$ be positive integers, let $({\mathbb C}^2)^{[n]}_{a,b}$ denote the Hilbert scheme of $n$ points on the plane fixed by the quasihomogeneous action with weights $(a,b)$, let $[\mathord{-}]$ denote its class in the relevant Grothendieck ring, let $t$ be a formal variable, and let ${\mathbb L}$ denote the class of the affine line. Generating-series conjecture. The generating series of these classes is

$$\sum_{n\ge 0}\left[({\mathbb C}^2)^{[n]}_{a,b}\right]t^n=\prod_{\substack{i\ge 1\\(a+b)\nmid i}}\frac{1}{1-t^i}\prod_{i\ge 1}\frac{1}{1-{\mathbb L}t^{(a+b)i}}.$$

The formula is based on computer calculations and predicts that the classes are governed only by the sum of the two weights. Its validity in general is left open by the source.