Every tournament contains a directed Hamiltonian path (a path visiting every vertex exactly once).

Suppose that $T$ is a tournament on $[n]:=\{1,\ldots,n\}$ for some integer $n\geq 2$ with exactly one directed Hamiltonian path. Does this imply that $T$ is isomorphic to the tournament $([n], E)$ where $E = \{(i,j)\in [n]: i<j\}$?