  Let $X$ be a convex polyomino such that its vertex set is a sublattice of
$\mathbb{N}^2$. Let $\Bbbk[X]$ be the toric ring (over a field $\Bbbk$)
associated to $X$ in the sense of Qureshi, \emph{J. Algebra}, 2012. Write the
Hilbert series of $\Bbbk[X]$ as $(1 + h_1 t + h_2 t^2 + \cdots
)/(1-t)^{\dim(\Bbbk[X])}$. For $k \in \mathbb{N}$, let $r_k$ be the number of
configurations in $X$ with $k$ pairwise non-attacking rooks. We show that $h_2
< r_2$ if $X$ is not a thin polyomino. This partially confirms a conjectured
characterization of thin polyominoes by Rinaldo and Romeo, \emph{J. Algebraic
Combin.}, 2021.
