Rational-span conjecture for tangent spaces of regular Hadamard matrices

Let $H\in M_N(\mathbb C)$ be a regular Hadamard matrix, and let $\widetilde{T}_H C_N$ denote the associated vector space of matrices. Write $[\widetilde{T}_H C_N]_{\mathbb Q}$ for the subset consisting of matrices $A\in\widetilde{T}_H C_N$ having rational entries.

Rational-span conjecture. One has

$$\widetilde{T}_H C_N=\mathbb C\cdot[\widetilde{T}_H C_N]_{\mathbb Q}.$$

The claim is known to hold for Fourier matrices; the conjecture is that it holds for every regular Hadamard matrix, with regularity understood in the sense used in the cited reference.