  We give an example of a pseudoconvex domain in a complex manifold whose
$L^2$-Dolbeault cohomology is non-Hausdorff, yet the domain is Stein. The
domain is a smoothly bounded Levi-flat domain in a two complex-dimensional
compact complex manifold. The domain is biholomorphic to a product domain in
$\mathbb{C}^2$, hence Stein. This implies that for $q>0$, the usual Dolbeault
cohomology with respect to smooth forms vanishes in degree $(p,q)$. But the
$L^2$-Cauchy-Riemann operator on the domain does not have closed range on
$(2,1)$-forms and consequently its $L^2$-Dolbeault cohomology is not Hausdorff.
