Sun's conjecture on reciprocal squared differences in a permutation

Let $n$ be a positive integer and let $\pi\in\mathfrak{S}_n$ be a permutation of $\{1,\ldots,n\}$. Sun's squared-difference conjecture. If $n>7$, there is a permutation $\pi$ such that

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

The source presents this as one of Sun's conjectures that it is not able to prove, concerning a rational function formed from consecutive squared permutation values.