OZF multiplier conjecture for bounded Lurye-system outputs

Let a Lurye system have a time-invariant nonlinearity $\boldsymbol{\phi}\in\Phi^{ti}$, with $r_1=0$ and $r_2\in\mathcal{L}_2\cap\mathcal{L}_{\infty}$. An OZF multiplier is a multiplier $\boldsymbol{M}\in\mathcal{M}$ suitable for $\boldsymbol{G}$, or for $1/k+\boldsymbol{G}$ when $\boldsymbol{\phi}\in\Phi^{sr}_k$ for some $k>0$.

OZF multiplier conjecture. If such a multiplier exists, then $y_2\in\mathcal{L}_2\cap\mathcal{L}_{\infty}$.

This is one of two component conjectures proposed as sufficient for the periodic-solution conjecture. The source does not report a resolution.