Conjecture on the monotonicity threshold and complete monotonicity of G_a

Let $R$ be the function defined in the source by equation (R(x)), and let $G_a$ be the parameterized function defined earlier in the paper. Monotonicity and complete-monotonicity conjecture. The function

$$x\mapsto \frac{-24\left(x^{4}-\frac{31}{336}\right)R(x)+x^{2}-\frac{7}{40}}{24\left(x^{2}+\frac{7}{40}\right)R(x)-1}$$

is increasing on $(0,\infty)$; moreover, $-G_a$ is completely monotone on $(0,\infty)$ if and only if $a\leq 155/294$. These are the two conjectures obtained from the preceding necessary condition and inequalities; the supplied text gives no resolution.