Divisibility conjecture for shifted Poincaré polynomials on $\mathbb P^2$

Let $d$ be a positive integer and let $\hat\Omega_d^{\mathbb P^2}$ be the shifted Poincaré polynomial associated with the moduli space of degree-$d$ one-dimensional sheaves on $\mathbb P^2$. Divisibility conjecture. If $3$ divides $d$, then

$$\frac{\hat\Omega_d^{\mathbb P^2}}{y^2+y+1}$$

is a palindromic polynomial in $y$. The conjecture is based on the paper's computations, and no proof or disproof is supplied.