Is it true that $$s_{n,k}:=\sum_{j=1}^{n-1} r_{n,k,j} <0$$ for all natural $n\ge2$ and all natural $k\in\{1,\dots,n-1\}$, where $$\text{$r_{n,k,j}:=\frac{x_{n,2j}}{y_{n,k,j}\;y_{n,k+1,j}},\quad$ $x_{n,j}:=\sin^2\frac{\pi j}{2 n},\quad$ and $\quad y_{n,k,j}:=2x_{n,j}-x_{n,k-1}-x_{n,k}$?}$$
This question was motivated by this previous one.
One may note here that for each natural $n\ge2$ and each natural $k\in\{1,\dots,n-1\}$ we have $r_{n,k,k}<0$, whereas $r_{n,k,j}>0$ for all $j\in\{1,\dots,n-1\}\setminus\{k\}$ – but the only negative summand $r_{n,k,k}$ in the sum $\sum_{j=1}^n r_{n,k,j}$ seems to more than counterbalance the other, positive summands $r_{n,k,j}$ with $j\ne k$. For instance, here are approximate values of $r_{10,4,j}$ for $j\in\{1,\dots,10-1\}$: $$0.24, 1.50, 11.00, -42.00, 14.00, 2.60, 0.85, 0.29, 0.06.$$
However, the counterbalancing effect seems somewhat delicate, as (at least for $k\in\{1,\dots,n-1\}$ far enough from $1$ and $n-1$) the ratio of the sum $s_{n,k}$ to $r_{n,k,k}$ seems to be decreasing to $0$ as $n$ increases to $\infty$.
