Loose Hamiltonian Cycles Forced by Large (k-2)-Degree - Approximate Version

We prove that for all $k\geq 4$ and $1\leq\ell<k/2$, every $k$-uniform hypergraph ${\mathcal{H}}$ on $n$ vertices with $\delta_{k-2}({\mathcal{H}})\geq(\tfrac{4(k-\ell)-1}{4(k-\ell)^2}+o(1))\binom{n}{2}$ contains a Hamiltonian $\ell$-cycle if $k-\ell$ divides $n$. This degree condition is asymptotically best possible. The case $k=3$ was addressed earlier by Bus et al.