Local transformations of superpositions of entangled states

Suppose that we have two entangled states $\ket {\phi_1}$, $\ket{\psi_1}$ that cannot be converted to any of other two states $\ket{\phi_2}$, $\ket{\psi_2}$ by local operations and classical communication. We analyze the possibility of locally transforming a superposition of $\ket{\phi_1}$ and $\ket{\psi_1}$ into a superposition of $\ket{\phi_2}$ and $\ket{\psi_2}$. By using the Nielsen's theorem we find the necessary and sufficient conditions for this conversion to be performed.