Empirical evidence suggests that, for each positive integer $n$, the following equality holds:
\begin{equation*} \prod_{s=1}^{2n}\sum_{k=1}^{2n}(-i)^k\sin\frac{sk\pi}{2n+1}=(-1)^n\frac{2n+1}{2^n}, \end{equation*} where $i=\sqrt{-1}$.

Is it a known equality? If it is true, would you please give me some insights on how to derive this equality?