TournamentClique completeness conjecture for the second level of the polynomial hierarchy

A tournament is a directed graph in which exactly one of the two possible directions is chosen for every pair of distinct vertices. For a tournament $T$, its directed clique number $\operatorname{\overrightarrow{\omega}}(T)$ is the minimum, over all linear orders $\ll$ on $V(T)$, of the clique number of the corresponding backedge graph $T^\ll$. The problem TournamentClique asks, given a tournament $T$ and a positive integer $t$, whether $\operatorname{\overrightarrow{\omega}}(T)\leq t$. The class $\Sigma^\mathsf{P}_2$ is the second level of the polynomial hierarchy. TournamentClique completeness conjecture. TournamentClique is $\Sigma^\mathsf{P}_2$-complete. This would settle the complexity of TournamentClique, which is already known to lie in $\Sigma^\mathsf{P}_2$; the conjectured hardness remains open.