Optimality conjecture for the natural orthogonal complement in hidden Markov model reduction

Let $cmathcal{E}_cperp$ be the natural orthogonal-complement effective subspace defined above, and let $coverline{\bm p}$ be the vector defined in Corollary corresponding to $cmathcal{E}_cperp$. For any other completion $cmathcal{E}$ and any non-negative vector $cbm w$, consider the reduced algebras obtained by the operation $\operatorname{alg}(\overline{\bm p}^{-1}\wedge\mathcal{E}_\perp)$ and $\operatorname{alg}(\bm w^{-1}\wedge\mathcal{E})$. Optimality conjecture. The natural orthogonal complement gives a reduced model of minimal dimension, namely

$$\dim(\mathcal{E}_\perp)\leq\dim(\operatorname{alg}(\overline{\bm p}^{-1}\wedge\mathcal{E}_\perp))\leq\dim(\operatorname{alg}(\bm w^{-1}\wedge\mathcal{E})).$$

This conjecture asserts that the natural orthogonal-complement choice is optimal among the possible effective subspaces and non-negative completion vectors, based on the analytical and numerical examples examined in the paper. Its resolution is not supplied in the given text.