Leading Betti-number expansion conjecture for moduli spaces on $\mathbb P^2$

For $d\geq 4$, let $M_d$ be the moduli space of stable one-dimensional sheaves of degree $d$ on $\mathbb P^2$, and let $\hat\Omega_d^{\mathbb P^2}$ denote its shifted Poincaré polynomial. Define

$$f(y)=\frac{(1+y+y^2)(-2+2y+4y^2+2y^3+y^4+2y^5)}{(1-y)(1-y^2)}.$$

Leading Betti-number expansion conjecture. One has

$$\left.\hat\Omega_d^{\mathbb P^2}\right|_{y^{\leq 3d-10}}=\left.\prod_{k>0}\frac{1}{(1-y^k)(1-y^{k+1})^2}\left(1-3y^{d-1}\frac{1+y+y^2}{1-y}+3y^{2d-4}f(y)\right)\right|_{y^{\leq 3d-10}}.$$

This extends the proven leading-Betti-number formula to a larger degree range and is supported by numerical calculations; no resolution is given in the source.