The 3-uniform tight-cycle Turán density conjecture

Let $C_s^3$ denote the $3$-uniform tight cycle of length $s$, and let $\operatorname{ex}_2(n,C_s^3)$ be the maximum number of edges in a $C_s^3$-free $3$-uniform hypergraph on $n$ vertices. Turán conjecture. For $s\not\equiv 0\bmod 3$,

$$\operatorname{ex}_2(n,C_s^3) = (1/3 + o(1))n.$$

The proposed lower bound comes from a construction using three disjoint vertex classes of size $n/3$. The asymptotic extremal value is not known in the stated cases.