Existence conjecture for a column permutation with no zero principal minors

Let $N$ be a positive integer, set $\omega=e^{2\pi i/N}$, and consider the Fourier matrix with permuted columns

$$\bigl(\omega^{k\sigma(\ell)}\bigr)_{0\leq k,\ell\leq N-1},$$

where $\sigma$ is a permutation of $\{0,1,\dots,N-1\}$. A zero principal minor is a principal minor whose determinant is zero.

Column-permutation conjecture. For every natural number $N$, there exists a permutation $\sigma$ such that the Fourier matrix with permuted columns has no zero principal minors.

This assertion concerns the possibility of arranging the Fourier columns so that every principal submatrix is invertible. The source presents it as a conjecture motivated by applications to Riesz bases, and provides no resolution.