This paper addresses the link between the characterization of spectral sets for orthogonal Fourier bases in \({\mathbb {Z}}_n^2\) and for discrete orthogonal Gabor bases in \({\mathbb {Z}}_n^2.\) For Fourier bases there is a well known conjecture (i.e., the Fuglede conjecture) which states that a set is spectral if and only if it is a tile. This conjecture has been disproved for \(d\ge 3\) (d for dimension) and remains open for \(d=1,2.\) A similar but stricter characterization for discrete orthogonal Gabor bases is conjectured here, which states that the support set shall not only be tiling but also either a subgroup of order n (i.e., a Lagrangian) or a tiling complement of such a subgroup. The additional requirement comes from restrictions on the window vector. The author has established this statement (in both directions) before for n being a prime number, the purpose of this paper is to extend this result to n being a prime square. As opposed to possible false first impressions, this is not a simple extension of the prime case, and actually relies heavily on several new techniques.
