The local-to-global conjecture for tournament clique number

For a tournament $T$, let $N^+(v)$ denote the out-neighbourhood of $v$, and let $\operatorname{\overrightarrow{\omega}}(T)$ denote the clique number. The local-to-global clique-number conjecture. There exists a function $g$ such that, for every integer $t$, if $T$ is a tournament such that for every $v\in V(T)$, $\operatorname{\overrightarrow{\omega}}(N^+(v))\leq t$, then $\operatorname{\overrightarrow{\omega}}(T)\leq g(t)$. This is presented as the clique-number analogue of a known local-to-global theorem for dichromatic number and remains open.