A sharp trigonometric double inequality

We prove that $$ \left(\frac{5-\sqrt{5}}{8}\right)^\frac{3}{2}+\frac{1}{2}\sin^3\frac{8\pi}{5}\leqslant\sum_{k=1}^{n}\frac{\sin^3 k\theta}{k}\leqslant 1\qquad\hbox{for all integers }n\geqslant 1\hbox{ and }\theta\in(0,\pi), $$ where both bounds are sharp. This gives an affirmative answer to a conjecture of Alzer and Koumandos.