The unique-norm integral Fourier matrix conjecture

Let $s$ be an integral Fourier matrix with dimensions $d_1,\ldots,d_n$, and suppose that all dimensions are equal:

$$d_1=\ldots=d_n.$$

Unique-norm integral Fourier matrix conjecture. Then

$$s\in\{\pm 1\}^{n\times n}.$$

The claim proposes that an integral Fourier matrix with a unique norm can have only $\pm 1$ as entries. The surrounding discussion presents this as an expectation that seems to hold in general, but provides no resolution.