  Gabor phase retrieval is the problem of reconstructing a signal from only the
magnitudes of its Gabor transform. Previous findings suggest a possible link
between unique solvability of the discrete problem (recovery from measurements
on a lattice) and stability of the continuous problem (recovery from
measurements on an open subset of $\mathbb{R}^2$). In this paper, we close this
gap by proving that such a link cannot be made. More precisely, we establish
the existence of functions which break uniqueness from samples without
affecting stability of the continuous problem. Furthermore, we prove the novel
result that counterexamples to unique recovery from samples are dense in
$L^2(\mathbb{R})$. Finally, we develop an intuitive argument on the connection
between directions of instability in phase retrieval and certain Laplacian
eigenfunctions associated to small eigenvalues.
