Rational defect conjecture for Butson matrices

Let $H\in C_N$ be a complex Hadamard matrix. Its defect is $d(H)$, and its rational defect is

$$d_{\mathbb Q}(H)=\dim_{\mathbb Q}\bigl(\widetilde{T}_H C_N\cap M_N(\mathbb Q)\bigr),$$

where $\widetilde{T}_H C_N$ is the enveloping tangent space at $H$. A matrix is a Butson matrix if all its entries are roots of unity of a common finite order.

Rational defect conjecture. For the Butson matrices we have $d_{\mathbb Q}(H)=d(H)$.

The rational defect measures the part of the enveloping tangent space defined over $\mathbb Q$, while the defect measures its full dimension. The statement proposes that these dimensions coincide for Butson matrices; the source gives no resolution.