Captain-cycle conjecture for minimal TEQ-retentive sets

Let $T$ be a tournament, and let $\mathfrak{Dom}(T)$ denote its directed domination graph. A set is TEQ-retentive when it satisfies the tournament-equilibrium-set retentiveness condition, and it is minimal if it has no proper TEQ-retentive subset. Captain-cycle conjecture. If $\mathfrak{Dom}(T)$ contains a directed cycle $C$, then $V(C)$ is the unique minimal TEQ-retentive set of $T$. The previously proved special case assumes $V(C)=V(T)$; this conjecture asks whether that assumption can be removed, and it would imply the size-three uniqueness conjecture.