A chain inequality conjecture for the Toader–Qi mean and classical means

Let $a,b>0$ with $a\neq b$. Denote by $A(a,b)$, $G(a,b)$, $L(a,b)$, and $\mathcal{I}(a,b)$ the arithmetic, geometric, logarithmic, and identric means, respectively, and by $TQ(a,b)$ the Toader–Qi mean. Toader–Qi mean chain conjecture. The inequalities

$$\sqrt{A(a,b)G(a,b)}<TQ(a,b)<\sqrt{L(a,b)\mathcal{I}(a,b)}<\frac{L(a,b)+\mathcal{I}(a,b)}{2}<\frac{A(a,b)+G(a,b)}{2}$$

hold. The conjecture is suggested by known bounds for the Toader–Qi mean and by an established chain of inequalities between the classical means; its status is not resolved in the supplied source.