Log-concavity and unimodality of Hodge numbers of Hilbert schemes of points over a surface

Let $S$ be a smooth projective complex surface with irregularity $q=h^{1,0}(S)$ and geometric genus $g=h^{2,0}(S)$, and let $S^{[n]}$ denote its Hilbert scheme of $n$ points. We prove that, for every $n\ge 0$, the sequence $${\left(h^{p,0}\right(S^{[n]}\left)\right)}_{p=0}^{2n}$$ is log-concave if and only if $g\le \left(\genfrac{}{}{0px}{}{q+1}{2}\right)$. Moreover, log-concavity of all these sequences is already equivalent to log-concavity of the sequence for $n=2$. We also prove that these sequences are unimodal for every $n\ge 0$ if and only if $q\ge 1$ or $g=0$, and that this condition is already detected by the sequence for $n=1$.