Feedback coherent-classical improvement conjecture

Let a coherent-classical estimation scheme with feedback be defined by the feedback plant, feedback coherent controller, coherent-classical homodyne detector, and coherent-classical estimator, with cost $\tilde{J}_c$. Let the corresponding purely-classical feedback estimation scheme have cost $\bar{J}_c$. For a homodyne angle $\theta$, write these costs as $\tilde{J}_c(\theta)$ and $\bar{J}_c(\theta)$, and let $\theta_{opt}$ be the best choice of homodyne angle.

Feedback coherent-classical improvement conjecture. If there exists a homodyne angle $\theta_i$ such that

$$\tilde{J}_c(\theta_i) \leq \bar{J}_c(\theta_i),$$

then

$$\tilde{J}_c(\theta_{opt}) \leq \bar{J}_c(\theta_{opt}).$$

This conjecture concerns whether any improvement from coherent-classical estimation with feedback persists at the best homodyne angle. It is proposed from numerical observations and is not proved in the source.