Procedure conjecture for the fixed point of $\hat{L}_{\pi}^{\dagger,0}$

Let $P_0,P_1,P_2,P_{11},P_{12},P_{21},P_{22}$ be the transition matrices defining the active pieces of $\hat{L}_{\pi}^{\dagger,0}$, let $x_0,x_1,x_2,x_{11},x_{12},x_{21},x_{22}$ be their respective fixed points, and let $J^*$ be the fixed point of $\hat{L}_{\pi}$. The feasible region is determined by $x_0\leq x_i\leq J^*$, together with membership of $x_i$ in the corresponding active piece. Procedure conjecture. The fixed point of $\hat{L}_{\pi}^{\dagger,0}$ can be found by calculating these fixed points, discarding any $x_i$ that fails either feasibility test, and then choosing whichever of $P_1$ or $P_2$ remains; if neither remains, choose the remaining fixed point whose elements have the highest sum. The source gives this as a proposed procedure based on the preceding two-state analysis, without establishing it as a theorem.