We know that, the speed of light in vacuum is independent of reference frame. I read the reason for this fact from Why and how is the speed of light in vacuum constant, i.e., independent of reference frame?. Similarly, is the speed of light in a medium of refractive index $n \neq 1$ also independent of reference frame?
I tried to frame my question visually. So, let us consider the following image:
The entire setup is in a medium of refractive index not equal to unity. The light source is at rest with respect to the medium. The violet coloured squares below the light beam numbered $1,2,$ and $3$ are speed detectors which we can use to determine the speed of light beam with respect to the reference frame attached to them. $1$ is at rest with respect to the light source. $2$ moves parallel to the light beam and towards the source with a speed of $v$ and $3$ moves away with the same speed. Now my question is, is the speed of light beam detected by the three detectors equal? If it's vacuum we know they are equal. But what happens in this case?
Image Courtesy: My Own Work :)