  Let $R$ be a commutative local ring with infinite residue field such that $2
\in R^{*}$. We prove that the natural homomorphism $H_{k}(O_{n,n}(R) ;
\mathbb{Z}) \rightarrow H_{k}(O_{n+1,n+1}(R); \mathbb{Z})$ is an isomorphism
for $k \leq n-1$ and surjective for $k \leq n$. This improves the range for
homological stability given by Mirzaii by 1 and generalises the analogous
result for fields obtained by Sprehn and Wahl to the case of local rings.
