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 $\delta_{k-2}(\mathcal{H})\geq\left(\frac{4(k-\ell)-1}{4(k-\ell)^2}+o(1)\right)\binom{n}{2}$ 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 Bu{\ss} et al.