Neither Ms. Perez nor Ms. Yanaka believes that watching as much television as her son Sam does will lead to anything productive.
Is her correct? or should it be their?
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Using her is ambiguous, as we have no way of determining which of the ladies Sam is the son of.
Using their is incorrect unless Sam is the son of both ladies.
The proper way to resolve the situation is to specify which one of them Sam is the son of, thus:
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"... her ..." is correct; their is not relevant.
The sentence makes sense in the context where it is referring to someone's son Sam, which someone would have been referred to already and is someone other than the two women named in the sentence. As in, maybe,