Polynomial-time conjecture for Hamiltonian cycles after adding fixed vertices

A semicomplete digraph is a digraph in which every pair of distinct vertices is joined by at least one directed arc. Fix an integer $k$. Consider digraphs obtained from a semicomplete digraph by adding $k$ new vertices and an arbitrary set of arcs involving the new vertices.

Conjecture on fixed-vertex extensions. For every fixed integer $k$, there exists a polynomial-time algorithm for deciding whether such a digraph has a Hamiltonian cycle.

The source states that this conjecture is open already for $k=2$. The resulting digraph need not be a split digraph because arcs may be added between the new vertices.