Cite this problem as Problem 5.
Among all density operators of two qubits with the same spectrum one may look for those maximizing some measure of entanglement. It turns out  that for the entanglement of formation, the relative entropy of entanglement and the negativity one gets the same maximally entangled states.
Is this true for arbitrary entanglement monotones?
Obvious variants of this problem are for higher dimensional systems and weaker constraints on the spectrum, e. g., largest eigenvalue or entropy.
(Refer to definitions of the measures of entanglement and `entanglement monotone’.)
 F. Verstraete, K. Audenaert, and B. De Moor, Maximally entangled mixed states of two qubits, quant-ph/0011110 (2000).