Let $X$ and $Y$ be regular integral Noetherian schemes. Assume that $X$ and $Y$ are smooth and proper over a base scheme $S=Spec R$, where $R$ is a discrete valuation ring.

If $X$ and $Y$ have isomorphic generic fibres, is it also the case that their special fibres are isomorphic?

Remarks:

The answer is yes when $X$ and $Y$ are abelian schemes (this follows from the theory of the Néron model). In general though it is not the case that morphisms between the generic fibres extend to the special fibre.

I am also particularly interested in case where $R$ is the localisation of $\mathbb{Z}$ at some prime, and hence the generic fibres are smooth proper varieties over $\mathbb{Q}$.