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

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