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It is not true that $\text{rank}_+^*(V) \leq \text{rank}_+(V) $. In fact, an equivalent definition of the non-negative rank of $V$ is the minimum number of non-negative vectors (not necessarily columns of $V$) such that every column of $V$ is a conic combination of these vectors. Therefore, the opposite inequality $\text{rank}_+^*(V) \geq \text{rank}_+(V) $ holds.