The sharp lower-bound conjecture for tight Hamilton cycles in Dirac hypergraphs

Fix an integer $k\geq 2$ and a constant $\gamma>0$. Let $G$ be an $n$-vertex $k$-graph with minimum codegree

$$\delta(G)\geq (1/2+\gamma)n.$$

A tight Hamilton cycle is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices forms an edge. Sharp lower-bound conjecture. The number of tight Hamilton cycles of $G$ is at least

$$(1/2-o(1))^n n!.$$

The paper proves the weaker order of magnitude $\exp(n\ln n-\Theta(n))$ under the same minimum-codegree condition and proposes this sharper asymptotic lower bound as a hypergraph analogue of a graph result. The supplied text gives no resolution status.