Do actors actually hire voice coaches to help RETAIN their accent?

At various points I heard that some non-native English speaking actors (noticably Arnold Schwarzenegger and Jackie Chan) could be speaking noticably better English, but in fact work with voice coaches to help retain their accents because to English speaking audiences their thick-accents are so recognizable the actors feel audiences wouldnt like them as much without their distinct accents.

Any truth to this?

I’m sure it’s true for some and not true for others.
And it does make sense. Even if they don’t lose their accent after living in the states for so long, they may still need a coach to teach them how to exaggerate theirs so they sound even more like themselves (does that make sense). For example, an American might be able to mimic an Australian accent, but could they go to Australia and make their American accent even more American? I wouldn’t even know where to start.