The quadratic-harmonic versus Seiffert mean conjecture

Let $x$ and $y$ be positive real numbers. Define the harmonic, geometric, arithmetic, and quadratic means by

$$H=\frac{2xy}{x+y},\qquad G=\sqrt{xy},\qquad A=\frac{x+y}{2},\qquad Q=\sqrt{\frac{x^2+y^2}{2}},$$

and define the Seiffert mean by

$$P=\frac{x-y}{2\arcsin\frac{x-y}{x+y}}\quad\text{if }x\ne y,\qquad P(x,x)=x.$$

Quadratic-harmonic versus Seiffert mean conjecture. For all positive real numbers $x$ and $y$,

$$Q+H\le 2P,$$

with equality if and only if $x=y$. This would strengthen the known inequality $A+G\le 2P$ in view of the proved inequality $A+G\le Q+H$; its status is left unresolved by the source.