The fundamental constants of the universe are not subject to change. That is true regardless of our ability to measure them accurately and precisely, and it would be true if we had no ability to measure them at all. Those constants existed long before we were able to measure them, and they will continue to exist long after our sun dims and life can no longer survive on earth.
The speed of light is a fundamental constant, and that article does not say otherwise. What that article says is not that the speed of light is variable, but that the density of subatomic particles may hinder the passage of light by causing it to enter and exit each particle during its passage through space. So if there are regions of space that contain more subatomic particles than other regions, the apparent speed of light will be slower there. The proposed effect of those subatomic particles is also so tiny that it will be very difficult to measure.
I will point out that the physics paper described in that article is a hypothetical paper, which means that its premise is consistent with existing theory and observation, but it has not yet been experimentally verified.
We already know that light interacts with matter; physicists established long ago that the apparent speed of light in air is less than in space, and its apparent speed in water is slower than in air. When light is absorbed by a particle, it becomes energy; when the energy is emitted by that particle, it becomes light. The speed of the light does not actually change; that apparent change is an artifact caused by the length of time light spends converted to energy within the particle.
The apparent speed of light is also affected by gravity. The stronger the gravity field is, the slower light goes. The only way to observe this is from a distance (for example, by looking at a black hole somewhere in the middle of the Milky Way). If you are subjected to the same gravitational field as the light, then relativistic effects cause the light speed you observe to remain at 300,000 km/sec--even if a distant observer sees the light traveling at only 250,000 km/sec in that same gravity field.
The fundamental flaw in science these days is that it forgets that it knows no FACT. It assumes, in it's hubris, to KNOW.
What science has discovered and ordained as 'fundamental constants' need not be 'fundamental' nor 'constant'.
As an example, simply tie time to entropy. If time is subject to entropy, and follows the standard rate of decay, then any equation assuming time is constant (the equation determining 'work' as an example) is highly flawed. Such an equation would seem to work perfectly today, and would seem to be highly provable... But applying the same to long ranging equations would cause them to become wildly inaccurate...
and while my little example here is eminently plausible, it is no more provable than any extrapolations performed by science, because the actual bonafide witnesses we have cannot be found beyond the advent of human records, and all that my premise would do in practice is bump the far end of time (the 'beginning') exponentially closer to now (in comparison to current models). In doing so, the 'change' could still be beyond our human witness.
Your statement above is exactly what I rise to oppose. Rigid constants may well be variables across the vast expanses being measured - Scientific extrapolations are so dramatically extrapolated without any assurance in their measurements, and that assumption can be fatal - Which is precisely why REAL science does not presume to 'know'.