  Entanglement distillation is a basic task in quantum information, and the
distillable entanglement of three bipartite reduced density matrices from a
tripartite pure state has been studied in [Phys. Rev. A 84, 012325 (2011)]. We
extend this result to tripartite mixed states by studying a conjectured matrix
inequality, namely $\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm
rank}(\sum_i R_i^T \otimes S_i)$ holds for any bipartite matrix $M=\sum_i R_i
\otimes S_i$ and Schmidt rank $K$. We prove that the conjecture holds for $M$
with $K=3$ and some special $M$ with arbitrary $K$.
