Polynomial Hamiltonian-cycle conjecture for split digraphs with bounded independent sets

A split digraph is a digraph $D=(V_1,V_2;A)$ whose vertex set is the disjoint union of an independent set $V_1$ and a set $V_2$ inducing a semicomplete digraph. A Hamiltonian cycle is a directed cycle containing every vertex exactly once.

Bounded-independent-set conjecture. For every fixed integer $k$, the Hamiltonian-cycle problem is polynomial-time solvable for the class of split digraphs $D=(V_1,V_2;A)$ in which the independent set satisfies $|V_1|\leq k$.

The conjecture proposes a polynomial algorithm for each fixed bound on the independent-set size. The source motivates it using the preceding fixed-vertex-extension conjecture, but gives no resolution evidence.