Li–Yorke sensitivity does not imply Li–Yorke chaos

We construct an infinite-dimensional compact metric space $X$ , which is a closed subset of $\mathbb{S}\times \mathbb{H}$ , where $\mathbb{S}$ is the unit circle and $\mathbb{H}$ is the Hilbert cube, and a skew-product map $F$ acting on $X$ such that $(X,F)$ is Li–Yorke sensitive but possesses at most countable scrambled sets. This disproves the conjecture of Akin and Kolyada that Li–Yorke sensitivity implies Li–Yorke chaos [Akin and Kolyada. Li–Yorke sensitivity. Nonlinearity 16, (2003), 1421–1433].