The codegree Turán density of $3$-uniform tight cycles

Given any $\epsilon >0$ we prove that every sufficiently large $n$-vertex $3$-graph $H$ where every pair of vertices is contained in at least $(1\mathrm{/}3+\epsilon )n$ edges contains a copy of $C_{10}$, i.e.\ the tight cycle on $10$ vertices. In fact we obtain the same conclusion for every cycle $C_{\mathrm{\ell }}$ with $\mathrm{\ell }\ge 19$.