The AVV conjecture on Ramanujan's gamma-function remainder

Let $\Gamma$ denote Euler's gamma function and define, for $x>0$, $G(x)$ and $H(x)$ by

$$G(x)=\left(\frac{e}{x}\right)^x\frac{\Gamma(1+x)}{\sqrt{\pi}}.$$ $$H(x)=G(x)^6-8x^3-4x^2-x=\frac{\theta_x}{30}.$$

AVV conjecture. The function $H$ is increasing from $(1,\infty)$ into $(1/100,1/30)$. This conjecture is attributed to the authors' earlier work and is motivated by Ramanujan's recorded approximation; its resolution is not given in the supplied text.