It is known to be true for all $n$ as a consequence of the stable homeomorphism conjecture (SHC), itself a consequence of the annulus conjecture.
The SHC says that any orientation preserving homeomorphism of $\mathbb{R}^n$ is a product of homeomorphisms each of which is the identity on some non-empty open set.
The same statement is then true for an homeomorphism $h$ of $\mathbb{S}^n$, since you can by a first homeomorphism assume that $h$ fixes the north pole. But an homeo of $\mathbb{S}^n$ which is the identity on a non-empty open ball is isotopic to the identity, by Alexander's trick.