Uniqueness conjecture for unmixed-symplectic spectral factors

Let $\Phi(z)\in\mathbb{R}(z)^{n\times n}$ be a spectrum of normal rank $\mathrm{rk}(\Phi)=r\neq 0$. Let $\mathscr{A}_p$ and $\mathscr{A}_z$ be two unmixed-symplectic sets. Let $W(z)$ be a spectral factor satisfying points 1, 2, and 3 of Theorem 3. Then the spectral-factor uniqueness conjecture. $W(z)$ is unique up to multiplication on the left by a constant orthogonal matrix: if $W_1(z)$ also satisfies points 1, 2, and 3 of Theorem 3, then

$$W_1(z)=T W(z),$$

where $T\in\mathbb{R}^{r\times r}$ is orthogonal. This conjecture concerns the link between arbitrary spectral factors and stochastic minimality and would provide a first step toward parametrizing all stochastically minimal right-invertible spectral factors.