Cyclic reciprocal squared-difference conjecture for permutations

Let $n$ be a positive integer and let $\pi\in\mathfrak{S}_n$ be a permutation of $\{1,\ldots,n\}$. Cyclic squared-difference conjecture. If $n>11$, there is a permutation $\pi$ such that

$$\sum_{k=1}^{n-1}\frac{1}{\pi(k)^2-\pi(k+1)^2}+\frac{1}{\pi(n)^2-\pi(1)^2}=0.$$

The paper introduces this as a further conjecture motivated by the preceding non-cyclic squared-difference conjecture, and explicitly distinguishes it from the conjectures of Sun it has just reproduced.