Hamiltonian prescribed-endpoint path under a cycle-factor condition

Prove that every 4-strong extended tournament with an $(x,y)$-path $P$ for which $D-V(P)$ has a cycle factor possesses a Hamiltonian $(x,y)$-path.

Background

This conjecture concerns a structural sufficient condition for extending a prescribed-endpoint path to a spanning path. The digraph is required to be a 4-strong extended tournament, and the vertices outside the given $(x,y)$-path must admit a cycle factor.

The paper's counterexample concerning the union of a prescribed-endpoint path and a disjoint directed cycle does not refute this conjecture because that example is neither 4-strong nor an extended tournament. The paper explicitly states that the conjecture remains unresolved in general.