A Hamiltonian path in a digraph $D$ in which the initial vertex dominates the terminal vertex is called a Hamiltonian bypass. Let $D$ be a 2-strong digraph of order $p\geq 3$ and let $z$ be some vertex of $D$. Suppose that every vertex of $D$ other than $z$ has degree at least $p$. We introduce and study a conjecture which claims that there exists a smallest integer $k$ such that if $d(z)\geq k$, then $D$ contains a Hamiltonian bypass. In this paper, we prove: (i) If $D$ is Hamiltonian or $z$ has a degree greater than $(p-1)/3$, then $D$ contains a Hamiltonian bypass. (ii) If a strong balanced bipartite digraph $B$ of order $2a\geq 6$ satisfies the condition that $d^+(u)+d^-(v)\geq a+1$ for all vertices $u$ and $v$ from different partite sets such that $B$ does not contain the arc $uv$, then $B$ contains a Hamiltonian bypass. Furthermore, the lower bound $a+1$ is sharp. The first result improves a result of Benhocine (J. of Graph Theory, 8, 1984) and a result of the author (Math. Problems of Computer Science, 54, 2020) We also suggest some conjectures and problems.