  Let $(X,J)$ be a $4$-dimensional compact almost-complex manifold and let $g$
be a Hermitian metric on $(X,J)$. Denote by
$\Delta_{\overline\partial}:=\overline\partial\overline\partial^*+\overline\partial^*\overline\partial$
the $\overline\partial$-Laplacian. If $g$ is globally conformally K\"ahler,
respectively (strictly) locally conformally K\"ahler, we prove that the
dimension of the space of $\overline\partial$-harmonic $(1,1)$-forms on $X$ is
a topological invariant given by $b_-+1$, respectively $b_-$. As an
application, we provide a one-parameter family of almost-Hermitian structures
on the Kodaira-Thurston manifold for which such a dimension is $b_-$. This
gives a positive answer to a question raised by T. Holt and W. Zhang.
