That's true and hasn't been addressed yet. But, i mean there is also the fact that Garam probably has a green card of sorts to work in the United states ( He works as a pianist). You can become a united states only after 5 years ( been 3 years already) if you have a permanent residence which is at Heejae place. Furthermore, a possibility of marriage with Heejae can officially make him a US citizen.

dropping my theory here so i can say i told ya so!! there is sooo much foreshadowing going on in season 5 that their secret relationship might be made public in season 6! hiding around on tour with compton, discussing their (controversial) age gap, the ghost of the manager still lingering (the author heavily implied the prof might‘ve been in love with the manager and maybe he pushed him away bc he couldn’t accept his feelings then), Garam fearing that the radio station caught wind of theor relationship bc they always play their performances in sequence, and in Ch 205 (mini spoiler) the professor’s former teacher askes him if they plan on keeping their relationship hidden.
dont know if it will hold up for a whole season worth of conflict/tension, but i think it might play a bigger part!
my dream ending for NME would be, that the professor goes full composer mode and keeps on writing music about his love for Garam. Garam keeps being a touring pianist and performs the prof‘s pieces in front of big audiences, therefore ultimately making their love public and letting the music and the performances speak for themselves.
it would also make Garam the ultimate Horowitz to the professor as Rachmaninoff.