OAMP state evolution conjecture

Let $\bm{A}$ be unitarily invariant, and let $\bm{W}_t$ be given by the specified linear estimator. Define

$$\tau _t^2 = \Phi _t\left(v_t^2\right),\qquad v_{t+1}^2=\Psi_t\left(\tau_t^2\right),$$

with $v_0^2=\operatorname{E}\{X^2\}$, where $\Phi_t$ and $\Psi_t$ are the state-evolution functions defined above. OAMP state evolution conjecture. The OAMP algorithm can be characterized by the state evolution recursion

$$\tau _t^2 = \Phi _t\left(v_t^2\right),\qquad v_{t+1}^2=\Psi_t\left(\tau_t^2\right).$$

This conjecture asserts that the scalar state evolution accurately describes OAMP under the stated unitary-invariance condition, extending the corresponding result for AMP; its resolution is not supplied here.