Chao-Ping Chen's upper Wilker inequality for inverse trigonometric functions

For $0<x<1$, the functions $\frac{\arcsin x}{x}$ and $\frac{\arctan x}{x}$ are defined, and the associated inequality is considered. Chao-Ping Chen's conjecture. If $0<x<1$, then

$$\left(\frac{\arcsin x}{x}\right)^2+\frac{\arctan x}{x}<2+\frac{\pi^2+\pi-8}{\pi}x^3\arctan x.$$

The constant $\frac{\pi^2+\pi-8}{\pi}$ should be the best possible. This is an upper counterpart to a previously proved lower inequality, and the proposed sharp constant is motivated by the limiting value of the corresponding quotient; the paper treats it as one of Chao-Ping Chen's open problems.