brauer groups – A local-to global principle for splitting of Azumaya algebras

Let $$S$$ be a finitely generated domain with the field of fractions $$F.$$ Let X be a smooth,
geometrically connected affine variety over $$S.$$ Let $$A$$ be an Azumaya algebra over $$X.$$
Assume that for all large enough primes $$p,$$ $$A_p$$ splits over $$X_p$$-the reduction modulo $$p$$ of $$X.$$ Does this assumption imply that $$A_{overline{F}}$$ splits over $$X_{overline{F}}?$$ My naive guess is that the answer should be “yes”. Any suggestions or references would be
greatly appreciated.