AFAIK inflation is an explanation for the way in which the early universe evolved into what we see. It does not emerge naturally from the standard model or general relativity. The possibility of a variable-over-time speed of light can offer alternative solutions to the horizon problem. However, there are several predictions which emerge from inflation theory and have been confirmed so as well as the proposed test variable speed would have to generate some explanation for those results. If not we could have two incompatible hypotheses and a lot of astrophysicists pulling their hair out.