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

Let $G$ be a directed $3$-uniform hypergraph in which every $3$-set supports at least four directed edges, and suppose that $G$ has no closed walk. A spanning path is a tight directed path containing every vertex exactly once.

Acyclic $(3,4)$-tournament spanning-path conjecture. If $G$ is a $(3,4)$-tournament with no closed walk, then $G$ has a spanning path.

This is presented as a particularly compelling open problem and as a strengthening target for the general $(3,4)$-tournament conjecture. Proving it would imply $f(n,3,4)\geq \Omega(n^{1/3})$, improving the cited lower bound.