One of the last open problems concerning two qubits in a pure state is to find the exact local content of their correlation, in the sense of Elitzur, Popescu, and Rohrlich (EPR2) [A. C. Elitzur, S. Popescu, and D. Rohrlich, Phys. Lett. A162, 25 (1992)]. We propose an EPR2 decomposition that allows us to prove, for a wide range of states |ψ(θ)〉=cosθ|00〉+sinθ|11〉, that their local content is pL̅ (θ)=cos2θ. We also share reflections on how to possibly extend this result to all two-qubit pure states.