Nonvanishing conjecture for principal submatrices of the Fourier matrix

Let $n$ be a square-free positive integer, let $\zeta_n$ be an $n$-th root of unity, and consider the matrix $F_n=(\zeta_n^{ij})_{0\leq i,j\leq n-1}$. A principal submatrix is a matrix $(\zeta_n^{ij})_{i,j\in I}$ for some subset $I\subseteq\{0,1,\ldots,n-1\}$. Nonvanishing conjecture. Every principal submatrix of $F_n$ has nonzero determinant.

This conjecture concerns the nonsingularity of principal submatrices of the discrete Fourier matrix in the square-free case. Singular non-principal submatrices are known when $n$ is not prime, but no counterexample to the principal-submatrix assertion is known; the conjecture was first stated by Cabrelli and presented in a more general form by Chebotarev and collaborators.