Rank of a matrix product

$A$ is a $4\times2$ matrix and $B$ is a $2\times4$ matrix. Their product $AB$ is a $4\times4$ matrix. What is the largest possible rank of $AB$ ?

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$\operatorname{rank}(AB)\le\min(\operatorname{rank}A,\operatorname{rank}B)$ — the product can be no richer than either factor.
A matrix's rank can't exceed its smaller dimension, so each factor's rank is capped at the inner size $2$ ; the $4\times4$ output shape is a red herring.

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