A proof of an open problem of Yusuke Nishizawa for a power-exponential function

This paper presents a proof of the following conjecture, stated by Nishizawa in [Appl. Math. Comput. 269, (2015), 146–154.]: for ${\displaystyle 0<x<\pi \mathrm{/}2}$ the inequality $ \displaystyle \frac{\sin{x}}{x} !>! \left(\frac{2}{\pi} + \frac{\pi!-!2}{\pi^{3}}(\pi^{2}!-!4x^{2})\right)^{\theta(x)}! $ holds, where $\displaystyle \theta(x) ! = ! -\frac{(48!-!24\pi!+!\pi^{3})x^{3} }{3(\pi!-!2)\pi^{3}}+\frac{\pi^{3}}{24(\pi!-!2)}.$