The codegree Turán density of tight cycles minus one edge

Given $\alpha >0$ and an integer $\mathrm{\ell }\ge 5$, we prove that every sufficiently large $3$-uniform hypergraph $H$ on $n$ vertices in which every two vertices are contained in at least $\alpha n$ edges contains a copy of $C_{\mathrm{\ell }}^{-}$, a tight cycle on $\mathrm{\ell }$ vertices minus one edge. This improves a previous result by Balogh, Clemen, and Lidický.