The Fejer-Jackson-Gronwall inequality involving the sine function is as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$

Here I ask the following related question.

QUESTION: Do we have $$\sum_{k=1}^n(-1)^k\left(\frac{\sin kx}k\right)^m<0<\sum_{k=1}^n\left(\frac{\sin kx}k\right)^m$$ for all $m,n=1,2,3,\ldots$ and $0<x<\pi$ ?

Actually I formulated this question in 2013. My numerical computation suggests that the answer should be positive. How to prove this?