Let $X$ be a irreducible regular projective variety over $Spec(O_K)$ for some number field $K$. Is it known that there exists at least one hypersurface over $Spec(O_K)$ such that cuts $X$ in a regular codimension 1 subvariety?

I think the full version of the Bertini theorem is not known and requires assumption of some conjectures including abc conjecture (At least over $\mathbb{Z}$). I was wondering whether the existence of just one such a hypersurface is known or not. Thanks!

hyperplanesection exists after base change to $O_L$. This follows from Rumely's existence theorem (and of course, it does not answer the question!) $\endgroup$ – Laurent Moret-Bailly Apr 15 at 9:56