No matter what they discovered: Einstein's theories of relativity will hold. They will hold forever, just like Newtons theory of gravity will hold forever. It's just that its range of applicability may become smaller and smaller.
There have been numerous experiments that perfectly validate both the special and general theory of relativity. The outcomes of the same experiments will not suddenly change. The theories predict the outcomes of these experiments to within experimental accuracy. GPS wouldn't work if special relativity wasn't a sufficiently accurate description of reality.
However, it may turn out that there are also experiments that are not sufficiently described by the theories of relativity, even thought that was previously expected.
There have been numerous experiments that perfectly validate both the special and general theory of relativity. The outcomes of the same experiments will not suddenly change. The theories predict the outcomes of these experiments to within experimental accuracy. GPS wouldn't work if special relativity wasn't a sufficiently accurate description of reality.
However, it may turn out that there are also experiments that are not sufficiently described by the theories of relativity, even thought that was previously expected.