No, like I said, the special theory of relativity is based on the speed of light being the same for all observers, not on its exact value. That is a supposition on which the theory is based, not one of its predictions.
If, for example we now find out that neutrinos move with 1.0002% of the speed of light, but that their speed is the same, measured by any observer, we will just replace the speed of light constant in the special relativity formulas with the speed of neutrinos constant.
But that all depends what the interpretations of the results are going to be.
