The combined parameterization conjecture for Hamiltonian Cycle

Let $G$ be a graph on $n$ vertices, and let $k$ be a nonnegative integer. Suppose that at least $n-k$ vertices of $G$ have degree at least $n/2-k$. Combined parameterization conjecture. The Hamiltonian Cycle problem with input $G$ can be solved in time

$$c^k \cdot n^{O(1)}$$

for some constant $c$. This would combine the paper's two parameterizations of Hamiltonicity below Dirac's condition; establishing such a fixed-parameter algorithm remains open.