Redundancy of the detectability condition for the optimal matrix pair

Let $(\Gamma,\Pi,\hat{\Sigma})$ be a solution of the convex optimization in, with matrices $F$, $G$, $H$, and $J$ as in Theorem. The matrix pair formed using the pseudoinverse $\hat{\Sigma}^\dagger$ is

$$(F+G\Gamma\hat{\Sigma}^\dagger,H+J\Gamma\hat{\Sigma}^\dagger).$$

A matrix pair is detectable when the associated linear system satisfies the detectability condition.

Detectability redundancy conjecture. The detectability condition in Theorem is redundant: if $(\Gamma,\Pi,\hat{\Sigma})$ solves the optimization in, then $(F+G\Gamma\hat{\Sigma}^\dagger,H+J\Gamma\hat{\Sigma}^\dagger)$ is detectable.

The detectability condition is required in the paper to establish achievability of the convex-optimization upper bound for the feedback capacity, although the condition can be checked through an LMI optimization. The conjecture is based on extensive simulations; the supplied text gives no proof or resolution.