Loose Hamiltonian cycles forced by large $(k-2)$-degree - sharp version

We prove for all $kgeq 4$ and $1leqell<k/2$ the sharp minimum $(k-2)$-degree bound for a $k$-uniform hypergraph $mathcal H$ on $n$ vertices to contain a Hamiltonian $ell$-cycle if $k-ell$ divides $n$ and $n$ is sufficiently large. This extends a result of Han and Zhao for $3$-uniform hypegraphs.