The $(3,4)$-tournament spanning-path conjecture

A $(3,4)$-tournament is a directed $3$-uniform hypergraph in which every $3$-set supports at least four directed edges. A spanning path is a tight directed path containing every vertex exactly once.

$(3,4)$-tournament spanning-path conjecture. Every $(3,4)$-tournament has a spanning path. That is, $f(n,3,4)=n$.

The conjecture would determine the maximum guaranteed spanning-path size for $(3,4)$-tournaments. The surrounding discussion states that current results only give a path of size $\mega(n^{1/5})$, so the spanning assertion remains open.