Kalman conjecture for absolute stability of feedback interconnections

Let $G$ be a stable transfer function, let $k_N$ be its Nyquist value, and let $\phi$ be a memoryless nonlinearity in $S[0,k]$. Kalman conjecture. The feedback interconnection of $G$ and $\phi$ is asymptotically stable for all memoryless $\phi\in S[0,k]$ if and only if $k<k_N$. This conjecture is refuted in continuous time by counterexamples, although it has been proved for first-, second-, and third-order systems.