  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 \emph{globally conformally
K\"ahler}, respectively \emph{(strictly) locally conformally K\"ahler}, we
prove that the dimension of the space of $\overline\partial$-harmonic
$(1,1)$-forms on $X$, denoted as $h^{1,1}_{\overline\partial}$, 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. Furthermore, the
previous example shows that $h^{1,1}_{\overline\partial}$ depends on the
metric, answering to a Kodaira and Spencer's problem. Notice that such
almost-complex manifolds admit both almost-K\"ahler and (strictly) locally
conformally K\"ahler metrics and this fact cannot occur on compact complex
manifolds.
