Orders that are étale-locally isomorphic

Let $ R$ be a semilocal Dedekind domain with fraction field $ F$. It is shown that two hereditary $ R$-orders in central simple $ F$-algebras that become isomorphic after tensoring with $ F$ and with some faithfully flat étale $ R$-algebra are isomorphic. On the other hand, this fails for hereditary orders with involution. The latter stands in contrast to a result of the first two authors, who proved this statement for Hermitian forms over hereditary $ R$-orders with involution. The results can be restated by means of étale cohomology and can be viewed as variations of the Grothendieck-Serre conjecture on principal homogeneous spaces of reductive group schemes. The relationship with Bruhat-Tits theory is also discussed.