Draganić–Keevash conjecture on Hamilton cycles after adding a random 2-factor

Let $G$ be an $n$-vertex $d$-regular graph, and let $F \sim G_{n,2}$ be a uniformly random 2-factor, that is, a spanning 2-regular graph, on the same vertex set. Draganić–Keevash conjecture. With high probability, $G\cup F$ is Hamiltonian. This conjecture extends the known result that adding a uniformly random 2-factor to a regular graph with minimum degree $\omega(\log^3 n)$ yields a Hamiltonian graph; the claim for arbitrary $d$-regular graphs remains open.