I am a non-native speaker.

Think about a man who is sitting beside his wife in a hospital all along. Would `unremitting` be unsuitable to describe him?

> The whole night through, the man sat unremittingly beside her.

I am asking because I looked up the definition for `unremitting` and found that both `relentless` and `incessant` are strong synonyms which I thought possess a rather negative connotation of rigidity, stubbornness or annoyance.