Near-Sidorenko conjecture for 3-uniform tight cycles

Let $C^{(3)}_{3\ell}$ denote the $3$-uniform tight cycle with $3\ell$ edges, and let $s(F)$ denote the Sidorenko exponent of a hypergraph $F$. Near-Sidorenko conjecture for 3-uniform tight cycles.

$$s\left(C^{(3)}_{3\ell}\right)=(3+o_{\ell}(1))\ell.$$

The paper states this as the conjectured asymptotically sharp upper-bound scale for $3$-partite $3$-uniform tight cycles; its preceding theorem gives an upper bound that is tight up to a multiplicative constant, while the exact asymptotic assertion remains open.