Zames-Falb multiplier equality conjecture for absolute stability

Let $G$ be an LTI system, let $k_N$ denote its Nyquist gain, and let $0<k<k_N$. Suppose there is no suitable LTI Zames-Falb multiplier for $G+1/k$. Zames-Falb multiplier equality conjecture. The Lurye system is not absolutely stable, and hence

$$k_{LTIZF}=k_{ZF}=k_{AS}.$$

This conjecture concerns whether the available LTI Zames-Falb multiplier bounds exactly characterize absolute stability; the source repeats it from earlier work, but provides no resolution here.