All-pass parametrization conjecture for minimal right-invertible 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 $W_-(z)$ be the spectral factor corresponding to Theorem 3, and let $\overline{W}_+(z)$ be the spectral factor corresponding to

$$\mathscr{A}_p=\mathscr{A}_z=\{z\in\mathbb{C}:|z|<1\}.$$

Define the all-pass function

$$T_0(z):=\overline{W}_+(z)W_-^{-R}(z).$$

All-pass parametrization conjecture. The set of all minimal right-invertible spectral factors of $\Phi(z)$ is

$$\left\{W(z)=T_1(z)W_-(z):\ T_1^\ast(z)T_1(z)=T_1(z)T_1^\ast(z)=I_r,\quad \delta_M(T_1(z))+\delta_M(T_0(z)T_1^\ast(z))=\delta_M(T_0(z))\right\}.$$

This proposed parametrization is intended to describe all stochastically minimal right-invertible spectral factors in terms of all-pass divisors of the generalized phase function $T_0(z)$.