Kühn, Lapinskas and Osthus's conjecture on Hamilton cycles and even factors

For a graph $G$, let $\operatorname{reg}_{\mathrm{even}}(G)$ denote the largest even degree of an even-regular spanning subgraph of $G$ (an even factor). Let $G$ be a Dirac graph, meaning a graph on $n$ vertices with minimum degree at least $n/2$.

Kühn--Lapinskas--Osthus conjecture. $G$ contains at least

$$\operatorname{reg}_{\mathrm{even}}(G)/2$$

edge-disjoint Hamilton cycles.

The paper proves an approximate asymptotic version, with a factor of $1-\varepsilon$, but does not state that the exact conjecture is resolved.