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Restriction on the rank of marginals of bipartite pure states

Consider a qubit-qutrit ($2 \times 3$) composite state space. Let $C(\{1}{2}I_2, \{1}{3}I_3)$ be a convex set of all possible states of composite system whose marginals are given by $\{1}{2}I_2$ and $\{1}{3}I_3$ in two and three dimensional spaces respectively. We prove that there exists no pure state in $C(\{1}{2}I_2, \{1}{3}I_3)$. Further we generalize this result to an arbitrary $m \times n$ bipartite systems. We prove that for $m < n$, no pure state exists in the convex set $C(ρ_A,ρ_B)$, for an arbitrary $ρ_A$ and rank of $ρ_B >m$.