Conjecture on Lyapunov representations of observable all-pass functions

Let $A \in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$ be given. An observable realization is a realization whose associated state-space pair is observable, and an all-pass function is a rational function satisfying the all-pass identity on the unit circle.

Lyapunov representation conjecture. There exists a symmetric matrix $P=P^{\top}$ satisfying

$$A P A^{\top}-P=BB^{\top}$$

if and only if there exist matrices $C\in\mathbb{R}^{m\times n}$ and $D\in\mathbb{R}^{m\times m}$ such that $Q(z)=C(zI-A)^{-1}B+D$ is an observable realization of an all-pass function.

The conjecture would remove the reachability assumption from the first part of the cited theorem, characterizing the relevant all-pass realizations through the discrete-time Lyapunov equation. The source indicates that the converse implication may be non-trivial; no resolution is supplied here.