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Could it be possible (as in strictly speaking, no evidence going against this) that light could be limiting towards a more fundamental speed, and that light truly does slow down when a mass goes a speed along with it, but that the difference in speed is so minuscule that it is near impossible to notice with instruments used today?

By the way, I do not consider this as a discussion, but rather a question with the answer of "possibly" or straight up "no", and evidence or reasoning supporting your claim. I will only choose the answer with the best supporting evidence or reasoning, not the "maybe" answer just because it makes me feel good.

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Yes it is possible, but the difference would need to be miniscule. Effectively your question reduces to one of the following: "does light have a truly nonzero rest mass?" and / or "is there a highly diffuse optical medium all around us which modern repetitions of the Michelson-Morley experiment have not yet detected?". Look up "Experimental Checks on Photon Mass" section in the Wikipedia Photon article. See also the "Recent Experiments" section in the Wikipedia Michelson-Morley experiment article.

Indeed, as far as I am aware, direct measurements of neutrino speeds have never yet shown a statistically significant difference between the measurements and the speed of light. This was certainly true of the infamous 2011 OPERA experiment, which ultimately established that the speed of light was within the bounds of experimental error. We only know that their speeds must differ from the universal cause-effect speed limit $c$ indirectly through the phenomenon of flavor oscillation, which implies that they must have nonzero rest mass of the order of $1{\rm eV}$ or less. This mass is so small that it is very hard to prepare something with such a small mass in such a low momentum state that it can be observed by our instruments. Incidentally, this mass is approximately the rest mass mass of an optical photon / exciton superposition that propagates in a optical medium as I calculate here. To understand more about what I mean when I say your question is equivalent to one of photon mass, you might find my answer here helpful.

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