Periodic-solution conjecture under uniform boundedness for 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 $T$. A solution is uniform-bounded and uniform ultimate-bounded when it has the corresponding boundedness properties stated in the source.

Periodic-solution conjecture. If all solutions are uniform-bounded and uniform ultimate-bounded, then there exists a nonzero periodic solution with period $T$, which need not be unique or attracting.

This is the second component conjecture proposed as sufficient for the broader Altshuller-multiplier claim. Its resolution is not given in the source.