  Let $\mathcal{P}$ be a simple thin polyomino and $\Bbbk$ a field. Let $R$ be
the toric $\Bbbk$-algebra associated to $\mathcal{P}$. Write the Hilbert series
of $R$ as $h_{R}(t)/(1-t)^{\dim(R)}$. We show that
$$(-1)^{\left\lfloor{\frac{\mathrm{deg} h_R(t)}{2}}\right\rfloor}h_{R}(-1) \geq
0$$ if $R$ is Gorenstein. This shows that the Gorenstein rings associated to
simple thin polyominoes satisfy the Charney-Davis conjecture.
