A note on the incompleteness of Afshari & Leigh's system Clo

The system $\mathsf{C}\mathsf{l}\mathsf{o}$ is a cyclic, cut-free proof system for the modal $\mu$-calculus. It was introduced by Afshari & Leigh as an intermediate system in their intent to show the completeness of Kozen’s axiomatisation for the modal $\mu$-calculus. We prove that $\mathsf{C}\mathsf{l}\mathsf{o}$ is incomplete by giving a valid sequent that is not provable in $\mathsf{C}\mathsf{l}\mathsf{o}$.