Sun's conjecture on reciprocal sums over adjacent permutation values

Let $n$ be a positive integer and let $\pi\in\mathfrak{S}_n$ be a permutation of $\{1,\ldots,n\}$. Sun's reciprocal-sum conjecture. If $n>6$, there is a permutation $\pi$ such that

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

Moreover, if $n>7$, there is a permutation $\pi$ such that

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

These assertions are among the conjectures of Sun that the paper explicitly says remain unproved; they ask for both a linear and cyclic permutation arrangement with prescribed reciprocal sum.