Multipartite Dirac conjecture for Hamiltonicity

For each integer $r\geq 4$, let $K_r$ be the complete graph on $r$ parts, let $\mathcal{C}_r$ denote the collection of Hamilton cycles on $r$ labelled vertices, and let $\pi_{K_r}(\mathcal{C}_r)$ be the $r$-partite density threshold for Hamiltonicity.

Multipartite Dirac conjecture. The Hamiltonicity threshold should converge to $\frac{1}{2}$ as $r\rightarrow\infty$:

$$\lim_{r\rightarrow\infty} \pi_{K_r}(\mathcal{C}_r)=\frac{1}{2}.$$

A theorem in the paper gives a lower bound strictly greater than $1/2$ for every fixed $r\geq 4$, while the conjecture predicts that these thresholds nevertheless approach $1/2$ asymptotically.