Universality-class extension conjecture for OAMP/VAMP

Let $\mathscr{U}$ be the universality class used in the paper, let $\mathscr{Q}$ be a matrix class, let OAMP/VAMP denote the stated algorithms, and let SE denote their state evolution. A matrix is spectrally convergent when its empirical spectral behavior has the convergence property used in the source, and let $\bm{\Xi}_{\rm PI}$ be any matrix defined in Theorem

. **Universality-class extension conjecture.** There \exists a broader matrix class $\mathscr{Q}$ such that

\mathscr{U}\subset\mathscr{Q},

and the error vectors in OAMP/VAMP are asymptotically IID Gaussian, so that the SE is accurate. Moreover, for any spectrally convergent $\bm{A}$ satisfying $\|\bm{A}\|_2\lesssim 1$, and any $\bm{\Xi}_{\rm PI}$ defined in Theorem

,

$$\bm{A}\bm{\Xi}_{\rm PI}\in\mathscr{Q}.$$

The conjecture would extend the paper’s universality framework beyond the currently verified class $\mathscr{U}$; the source motivates it by numerical agreement between OAMP/VAMP performance and state evolution, but provides no proof.