Lemma: Product-free range is a sufficient witness

Every nonzero separable positive matrix has a nonzero product vector in its range. Hence a nonzero positive matrix with product-free range is entangled.

Proof. Write a separable positive matrix as $Z={\sum }_{\alpha }|u_{\alpha }\otimes v_{\alpha }⟩⟨u_{\alpha }\otimes v_{\alpha }|$. Its kernel is the intersection of the orthogonal complements of these product vectors, so every summand vector lies in $(\ker Z{)}^{\perp }=\mathrm{ran}Z$. At least one is nonzero.