Refined Joyce conjecture for one-dimensional sheaves on $\mathbb P^2$

For $d\in\mathbb Z_{>0}$ and $\chi\in\mathbb Z$, let $M_{d,\chi}$ be the moduli space of one-dimensional Gieseker semistable sheaves, and let $\Omega_{d,\chi}^{\mathbb P^2}(y^{\frac12})$ be the refined invariant formed from the even intersection Betti numbers $Ib_{2j}(M_{d,\chi})$. Here $y$ is a formal variable. Refined Joyce conjecture. For every $d\in\mathbb Z_{>0}$ and every $\chi,\chi'\in\mathbb Z$, one has

$$\Omega_{d,\chi}^{\mathbb P^2}(y^{\frac12})=\Omega_{d,\chi'}^{\mathbb P^2}(y^{\frac12}).$$

Thus the refined invariant, and equivalently the stated intersection-Betti-number data, should be independent of the Euler-characteristic parameter. The source identifies this as a refined version of a conjecture on Donaldson–Thomas counts; no resolution is supplied in the given text.