Iommi–Kiwi conjecture on inflections of two-branch Lyapunov spectra

Let $f$ be a two-branch expanding map, and let $L$ denote its Lyapunov spectrum. Iommi–Kiwi conjecture. The Lyapunov spectrum $L$ has at most two points of inflection. This conjecture concerns the possible shapes of Lyapunov spectra for finite-branch expanding maps. Earlier examples with non-concave spectra have precisely two inflection points, while the general upper bound remains open.