Bounded Sidorenko exponents for higher-uniformity tight cycles

For each integer $r\geq 4$, let $C^{(r)}_{r\ell}$ denote the $r$-uniform tight cycle with $r\ell$ edges, and let $s(F)$ denote the Sidorenko exponent of a hypergraph $F$. Conjecture on higher-uniformity tight cycles. For all $r\geq 4$, there is a constant $K_r$ depending only on $r$ such that, for every $\ell\geq 2$,

$$s\left(C^{(r)}_{r\ell}\right)\leq K_r r\ell.$$

This proposes the higher-uniformity analogue of the paper's upper bound for $3$-uniform tight cycles. The authors explicitly say that they could not find reasonable upper bounds in these cases, so the assertion remains open.