  Let $r \geq 2$ be an integer, and let $a$ be an integer coprime to $r$. We
show that if $c_2 \geq n + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 +
1) \right\rfloor$, then the $2n$th Betti number of the moduli space
$M_{\mathbb{P}^2,H}(r,aH,c_2)$ stabilizes, where $H =
c_1(\mathcal{O}_{\mathbb{P}^2}(1))$.
