The phrase "becomes engaged" implies the act of committing or involving oneself in a relationship or an activity. The antonyms for "becomes engaged" would be words that suggest detachment, disinterest, or neglect. Some possible antonyms may include "disengages," "dissociates," "abandons," "dismisses," "avoids," or "ignores." Alternatively, antonyms could refer to a lack of progress or advancement, such as "stagnates," "regresses," "fails," or "flounders." Ultimately, the choice of an antonym depends on the context and the intended meaning of the original phrase.