The Gaussian square-free walk-matrix conjecture

Let $G\in\mathcal{G}_n^{sc}$ be a self-converse mixed graph, let $W=W(G)$ be its walk-matrix, let $\mathscr{U}_G$ be the associated set of Gaussian rational unitary matrices, and let $\ell(U)$ denote the Gaussian integer scale parameter used in the paper. Gaussian square-free walk-matrix conjecture. If

$$\frac{\det W}{2^{\lfloor n/2\rfloor}}$$

is square-free in $\mathbb{Z}[i]$, then for every $U\in\mathscr{U}_G$, $\ell(U)=1$. The paper states this as an equivalent Gaussian-integer formulation of the generalized spectral determination conjecture; its resolution is not supplied in the source.