Turán density of tight cycles minus one edge in the ${\mathrm{\ell }}_2$-norm

The $3$-uniform tight $\mathrm{\ell }$-cycle minus one edge $C_{\mathrm{\ell }}^{3-}$ is the $3$-graph on $\mathrm{\ell }$ vertices consisting of $\mathrm{\ell }-1$ consecutive triples in the cyclic order. We show that for every integer $\mathrm{\ell }\ge 5$ satisfying $\mathrm{\ell }\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}3)$, every $C_{\mathrm{\ell }}^{3-}$-free $3$-graph whose ${\mathrm{\ell }}_2$-norm, that is, the sum of codegree squares, is close to the maximum must be structurally close to the iterative blowup of a single triple. This confirms a conjecture of Balogh–Clemen–Lidický~[Surveys in combinatorics 2022, 21-63] in a stronger form.