We know for a pure state conversion $|\psi \rangle \rightarrow_\textrm{LOCC} |\phi \rangle$ via local operation and classical communication (LOCC), an entanglement monotone should not increase, that is, $E(|\psi \rangle) \geq E(|\phi \rangle)$.
Say for a pure to mixed state conversion $|\psi \rangle \rightarrow_\textrm{LOCC} \rho$ given by the unique ensemble $l = \{ q_l, |\xi_l \rangle \} \in \mathcal{D}$ fixed by pointer states (which are omitted), where $\mathcal{D}$ is the set of all decompositions for $\rho$.
Naturally, the convex roof extension $E_\textrm{min}(\rho) = \min_\mathcal{D} \Sigma_i q_i |\xi_i \rangle \langle \xi_i|$ should not increase, nor should that of the obtained ensemble $l$ on average, i.e., $E(|\psi \rangle) \geq E_l(\rho) \geq E_\textrm{min}(\rho)$.
So, is it for certain there is not some decomposition $j = \{ q_j, |\xi_j \rangle \}$ that violates the nonincreasing criterion for $E$ on average, i.e., $E_j(|\rho \rangle) \geq E(\psi \rangle)$?