Rödl–Ruciński conjecture relating Hamilton cycles and perfect matchings

Let $k\ge3$ and $1\le d\le k-2$. For an $n$-vertex $k$-uniform hypergraph, let $h_d(k,n)=h_d^{k-1}(k,n)$ be the smallest integer such that minimum $d$-degree at least $h_d(k,n)$ guarantees a Hamilton tight cycle, and let $m_d(k,n)$ be the smallest integer such that minimum $d$-degree at least $m_d(k,n)$ guarantees a perfect matching. Rödl–Ruciński conjecture.

$$h_d(k,n)=m_d(k,n)+o(n^{k-d}).$$

This predicts that the asymptotic minimum $d$-degree thresholds for Hamilton tight cycles and perfect matchings coincide. The source presents it as an open conjecture and discusses known results for the case $d=k-1$.