Toader–Qi mean versus the 3/2-order logarithmic mean conjecture

Let $a,b>0$ with $a\neq b$. The $p$-order logarithmic mean is defined by

$$L_p(a,b)=L\left(a^p,b^p\right)^{1/p}.$$

Toader–Qi/logarithmic-mean conjecture. The inequality

$$TQ(a,b)>L_{3/2}(a,b)$$

holds. The conjecture is motivated by the known inequality $L(a,b)^{3/4}A(a,b)^{1/4}<L_{3/2}(a,b)$ and the corresponding established lower bound for the Toader–Qi mean; its status is not resolved in the supplied source.