Conjecture on near-optimal Hamilton coverings of random graphs

Let $G(n,p)$ be the binomial random graph on $n$ vertices, and let a Hamilton covering be a collection of Hamilton cycles whose union contains every edge of $G(n,p)$. Here $p=p(n)$ may depend on $n$, and $p=\omega(\ln n/n)$ means that $p n/\ln n\to\infty$.

Hamilton covering conjecture. For any $p=\omega(\ln n)/n$, the random graph $G(n,p)$ a.a.s. admits a covering of its edges with at most

$$(1+o(1))\frac{np}{2}$$

Hamilton cycles.

This would extend the paper's asymptotic equality between the largest Hamilton packing and smallest Hamilton covering to the full range where the random graph's minimum and maximum degrees are asymptotically equal. The authors state that the stronger lower bound used in their proof is likely only a proof artifact, and that this claim should hold whenever $p=\omega(\ln n)/n$; no resolution is given.