Sidorenko-gap conjecture for odd linear cycles

For an $r$-uniform linear cycle $C^r_{2\ell+1}$ of odd length $2\ell+1$, let $s(C^r_{2\ell+1})$ denote its Sidorenko gap. Odd linear-cycle Sidorenko-gap conjecture. For $r\geq 3$ and $\ell\geq 2$,

$$s(C^r_{2\ell+1})=\frac{1}{(r-1)\ell-1}.$$

The paper proves an upper bound on this gap and gives a conditional lower bound, so the asserted equality remains open and would require a matching upper bound.