The exponent-range conjecture for the four classical means

Let $x$ and $y$ be positive real numbers, and let

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

For a real number $n$, consider the inequality

$$A^n+G^n\le Q^n+H^n.$$

Exponent-range conjecture for the four classical means. The inequality holds for every negative real number $n$ and every real number $n\geq\frac12$. Moreover, for every real number $n\in(0,\frac12)$, neither this inequality nor its converse holds universally for all positive real numbers $x$ and $y$. The conjecture asks for the complete real-exponent range of validity of the inequality; the source provides computational and graphical evidence but no resolution.