Hamiltonicity threshold conjecture via linear forests

Let $n$ be even, let $\alpha\in(0,1/2)$ be fixed, and set $d=\alpha n$. Let $H_n$ be the complete bipartite graph with parts of sizes $d$ and $n-d$.

Hamiltonicity threshold conjecture. The sharp $d$-threshold for $H_n\cup G(n,p)$ to be Hamiltonian coincides with the sharp threshold for $G(n-d,p)$ to contain a linear forest with $n-2d$ edges.

A linear forest in the random graph on the larger part can be incorporated into a Hamilton cycle of the perturbed complete bipartite graph. The paper does not give an explicit threshold because the size of the largest linear forest in a sparse random graph has not been determined in the literature.