Relative Turán density of ordered cycles

Determine the relative Turán density \(\mathfrak{o}(C_\ell)\) of the ordered cycle \(C_\ell\) for every fixed \(\ell\ge 3\).

Background

The ordered cycle $C_\ell$ has vertex set $[\ell]$ and contains the monotone path $P_{\ell-1}$. Theorem 1.2 therefore gives the lower bound $\mathfrak{o}(C_\ell)\ge (\ell-2)/(2\ell-2)$, but the exact relative Turán density is not determined in the paper.