I think its kinda like Heejae branding his students, like his marking those who are going to be his chosen ones. In ch 3 other than naeun, the other 3 students don't have the mark like garam and he's only been there for a short period of time. Naeuns probably also wasn't chosen since shes talented but not unique like garam. She doesn't have that thing that heejae is looking for, to the point that she has to copy garam's sound. Butttt it also could be because shes a nepo baby and the niece from the woman heejae works for, so he can't really harm her or he might get punished for it.

Naeun never had a scar from the professor unlike Garam, now that we know the professor’s students all have the exact same scar (the one under their eye presumably from the professor throwing the music sheets resulting in a paper cut?) does that symbolizes that the professor never once acknowledged her as his student? idk just something I noticed