We have $$\sin\frac{\pi}{9}+\sin\frac{2\pi}9-\sin\frac{4\pi}9=\sin\frac{2\pi}{18}+\sin\frac{4\pi}{18}-\sin\frac{8\pi}{18}=\sin\frac{2\pi}{18}-\frac{8\pi}{18}+\sin\frac{14\pi}{18},$$ and the latter, denoting $\xi_{18}=\exp\frac{2\pi i}{18}$, is the imaginary part of $$\xi_{18}-\xi_{18}^4+\xi_{18}^7=\xi_{18}(1-\xi_{18}^3+\xi_{18}^6)=0.$$ Thus, your conjecture is wrong.

We have $$\sin\frac{\pi}{9}+\sin\frac{2\pi}9-\sin\frac{4\pi}9=\sin\frac{2\pi}{18}+\sin\frac{4\pi}{18}-\sin\frac{8\pi}{18}=\sin\frac{2\pi}{18}-\sin\frac{8\pi}{18}+\sin\frac{14\pi}{18},$$ and the latter, denoting $\xi_{18}=\exp\frac{2\pi i}{18}$, is the imaginary part of $$\xi_{18}-\xi_{18}^4+\xi_{18}^7=\xi_{18}(1-\xi_{18}^3+\xi_{18}^6)=0.$$ Thus, your conjecture is wrong.

We have $$\sin\frac{\pi}{9}+\sin\frac{2\pi}9-\sin\frac{4\pi}9=\sin\frac{2\pi}{18}+\sin\frac{4\pi}{18}-\frac{8\pi}{18}=\sin\frac{2\pi}{18}-\frac{8\pi}{18}+\sin\frac{14\pi}{18},$$ and the latter, denoting $\xi_{18}=\exp\frac{2\pi i}{18}$, is the imaginary part of $$\xi_{18}-\xi_{18}^4+\xi_{18}^7=\xi_{18}(1-\xi_{18}^3+\xi_{18}^6)=0.$$ Thus, your conjecture is wrong.

We have $$\sin\frac{\pi}{9}+\sin\frac{2\pi}9-\sin\frac{4\pi}9=\sin\frac{2\pi}{18}+\sin\frac{4\pi}{18}-\sin\frac{8\pi}{18}=\sin\frac{2\pi}{18}-\frac{8\pi}{18}+\sin\frac{14\pi}{18},$$ and the latter, denoting $\xi_{18}=\exp\frac{2\pi i}{18}$, is the imaginary part of $$\xi_{18}-\xi_{18}^4+\xi_{18}^7=\xi_{18}(1-\xi_{18}^3+\xi_{18}^6)=0.$$ Thus, your conjecture is wrong.