The short answer is we simply do not know: this hypothesis is utterly beyond anything that we can either test experimentally or reason about with a widely accepted theory.
Симон Тыран's Answer is a good, concise exposition showing what classical, relativistic reasoning has to say about this. But I don't believe this answers the question because you have to make the assumption that classical relativity works down to an arbitrarily small scale and that's something we neither know nor (I get the impression as a lay reader) even believe.
Moreover, even without full quantum gravity, one can formulate classical theories with both an invariant velocity (as in the $c$ of special relativity) and and invariant length scale. Examples of such theories are "Doubly Special Relativity" and also de Sitter Invariant Special Relativity wherein the symmetry group $SO(1,\,4)$ is a supergroup of the Lorentz group and is the same as the symmetry group of de Sitter space, a highly symmetric vacuum solution of the Einstein field equations. In such a universe one would have a natural, invariant length scale that can be used to invalidate the classical special relativistic reasoning that there always exists an inertial observer whom a wavelength is arbitrarily small for.
