The regular-subgraph conjecture for Hamilton-cycle packings

Let $G$ be a graph on $n$ vertices, and let $\operatorname{reg}_{\operatorname{even}}(G)$ denote the largest degree of an even-regular spanning subgraph of $G$. The regular-subgraph conjecture. If

$$\delta(G)\ge \frac{n}{2},$$

then $G$ contains at least

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

edge-disjoint Hamilton cycles. This graph-by-graph strengthening of the preceding conjecture is best possible for each individual graph. It was proved when $\delta\ge(2-\sqrt{2}+\varepsilon)n$, but remains open in general.