In their present, ether-free, form Maxwell's equations are in no conflict with Galilean relativity. The problem is vacuous but is restated again and again because there is a constant need to justify the introduction of Einstein's 1905 (false) constant-speed-of-light postulate. There are two major "justifications":
Maxwell's 19th century electromagnetic theory allegedly predicted that the speed of light does not depend on the speed of the observer measuring it. The truth is that Maxwell's theory predicted that the speed of light VARIES with the speed of the observer.
The Michelson-Morley experiment allegedly showed that the speed of light does not depend on the speed of the light source. The truth is that in 1887 the Michelson-Morley experiment unequivocally showed that the speed of light DOES DEPEND on the speed of the light source, as predicted by Newton's emission theory of light:
John Norton: "These efforts were long misled by an exaggeration of the importance of one experiment, the Michelson-Morley experiment, even though Einstein later had trouble recalling if he even knew of the experiment prior to his 1905 paper. This one experiment, in isolation, has little force. Its null result happened to be fully compatible with Newton's own emission theory of light. Located in the context of late 19th century electrodynamics when ether-based, wave theories of light predominated, however, it presented a serious problem that exercised the greatest theoretician of the day."
"Relativity and Its Roots", Banesh Hoffmann, p.92: "Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether.
John Norton: "In addition to his work as editor of the Einstein papers in finding source material, Stachel assembled the many small clues that reveal Einstein's serious consideration of an emission theory of light; and he gave us the crucial insight that Einstein regarded the Michelson-Morley experiment as evidence for the principle of relativity, whereas later writers almost universally use it as support for the light postulate of special relativity. Even today, this point needs emphasis. The Michelson-Morley experiment is fully compatible with an emission theory of light that CONTRADICTS THE LIGHT POSTULATE."