Nonvanishing principal-minor conjecture for Fourier matrices of square-free order

Let $N$ be a positive square-free integer, and let ${\mathcal{F}}_N=({\omega}^{ij})_{0\leq i,j\leq N-1}$ be the Fourier matrix, where ${\omega}=e^{-2\pi i/N}$ is a primitive $N$-th root of unity. A principal minor is the determinant of a square submatrix obtained by selecting the same index set for its rows and columns.

Square-free Fourier principal-minor conjecture. If $N$ is square-free, then all principal minors of ${\mathcal{F}}_N$ are nonzero.

This extends the nonvanishing phenomenon known for Fourier matrices of prime order and asks for it for every square-free order. The supplied source does not state whether the conjecture has been resolved.