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][1]. 

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][2].


  [1]: http://en.wikipedia.org/wiki/Annulus_theorem
  [2]: http://en.wikipedia.org/wiki/Alexander%2527s_trick