  For two given full-rank lattices $\mathcal{L}=A\mathbb{Z}^d$ and
$\mathcal{K}=B\mathbb{Z}^d$ in $\mathbf{R}^d$, where $A$ and $B$ are
nonsingular real $d\times d$ matrices, a function $g(\bf{t})\in
L^2(\mathbf{R}^d)$ is called a Parseval Gabor frame generator if
$\sum_{\bf{l},\bf{k}\in\mathbb{Z}^d}|\langle f, {e^{2\pi i\langle
B\bf{k},\bf{t}\rangle}}g(\bf{t}-A\bf{l})\rangle|^2=\|f\|^2$ holds for any
$f(\bf{t})\in L^2(\mathbf{R}^d)$. It is known that Parseval Gabor frame
generators exist if and only if $|\det(AB)|\le 1$. A function $h\in
L^{\infty}(\mathbf{R}^d)$ is called a functional Gabor frame multiplier if it
has the property that $hg$ is a Parseval Gabor frame generator for
$L^2(\mathbf{R}^d)$ whenever $g$ is. It is conjectured that an if and only if
condition for a function $h\in L^{\infty}(\mathbf{R}^d)$ to be a functional
Gabor frame multiplier is that $h$ must be unimodular and
$h(\bf{x})\overline{h(\bf{x}-(B^T)^{-1}\bf{k})}=h(\bf{x}-A\bf{l})\overline{h(\bf{x}-A\bf{l}-(B^T)^{-1}\bf{k})},\
\forall\ \bf{x}\in \mathbf{R}^d$ {\em a.e.} for any $\bf{l},\bf{k}\in
\mathbb{Z}^d$, $\bf{k}\not=\bf{0}$. The if part of this conjecture is true and
can be proven easily, however the only if part of the conjecture has only been
proven in the one dimensional case to this date. In this paper we prove that
the only if part of the conjecture holds in the two dimensional case.
