In Arkham Horror: The Card Game LCG, after an investigator evades a monster that is engaged with them, does the monster immediately become exhausted?

In Arkham Horror: The Card Game: an enemy is engaged with me. I perform an Evade action and pass the agility test. Does this enemy immediately become exhausted and disengage, remaining at the current location? During the subsequent enemy phase, will it still attack me?

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