Optimal-angle no-improvement conjecture for coherent-classical estimation

Let a coherent-classical estimation scheme and its corresponding purely-classical estimation scheme use the same plant, with costs $\tilde{J}_c$ and $\bar{J}_c$, respectively. Let $\theta_{opt}$ denote the optimal homodyne angle.

Optimal-angle no-improvement conjecture. At the optimal homodyne angle,

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

The conjecture says that, even when coherent controllers with squeezing are allowed, the purely-classical estimator performs at least as well at the best homodyne angle. The source presents this as an empirical observation and leaves it unproved.