The Balian–Low theorem for symplectic lattices in higher dimensions

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Linear Independence of Finite Gabor Systems (2006, background): “In particular, it is shown in [59] that the Classical BLT generalizes to the case of symplectic lattices in higher dimensions, but for nonsymplectic lattices little is known (see also Section 9.”
The Mystery of Gabor Frames (2014, unknown): “This is the so-called BalianLow-Theorem; see [7,13] for two surveys and [27,41] for the multivariate case.”
Density, Overcompleteness, and Localization of Frames. II. Gabor Systems (2005, result): “Some recent results on the BLT in higher dimensions are in [BCGP02], [GHHK02].”
An Introduction to Weighted Wiener Amalgams (2003, background): “We refer to [11] for discussion, history, proofs, and references, to [10] for recent results on the sharpness of the BLT, and to [40] for issues in higher-dimensional versions of the BLT.”
Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator (2004, unknown): “Non-uniform Gabor frames are not yet fully understood, some results can be found in [8, 12, 32, 34 , 41].”