They must stay true to their personality or they are lifeless figurines doing a meaningless dance.
I think a more fitting expression would be they have to stay true to their character. There's a difference between characterisation and true character.
Characterisation consists of personality, traits, quirks. It is how a character acts under normal circumstances. A construct. The mask we wear to fool ourselves and those we interact with into believing whatever it is we want them/ourselves to believe about us.
True character is how someone acts when the shit hits the fan. A soldier may believe himself a coward (characterisation), but when the bullets start flying somehow he gets up and leads a charge (character). Or the other way around–he may act like a gruff, battle-hardened veteran, but when the shells start falling he curls up in a ball and cries for his mother.
Great stories often put characterisation and character at odds to give us complex protagonists. Example: Walter White, a man who does horrible things and justifies them by claiming he has the good of his family at heart, while in fact the only interests he serves are his own pride and greed.