Altshuller multiplier conjecture for periodic solutions of Lurye systems

Let a Lurye system have a time-invariant nonlinearity $\boldsymbol{\phi}\in\Phi^{ti}$, with $r_1=0$ and a nonzero periodic input $r_2$ of period $T>0$. An Altshuller multiplier is a multiplier $\boldsymbol{M}\in\mathcal{A}_T$ suitable for $\boldsymbol{G}$, or for $1/k+\boldsymbol{G}$ when $\boldsymbol{\phi}\in\Phi^{sr}_k$ for some $k>0$.

Altshuller multiplier conjecture. If such a multiplier exists, then there exists a nonzero periodic solution with period $T$, which need not be unique or attracting.

The conjecture seeks a general periodic-response result for Lurye systems under periodic excitation. The surrounding results establish related conclusions under stronger hypotheses, but the proposed general statement is presented as unresolved.