Conjecture on the sign of the Stolarsky and power-mean difference

Let $L_\alpha$ denote the Stolarsky mean, let $B_p$ and $B_q$ be power means, and let $\mathcal{R}(B_p,L_\alpha,B_q)$ denote their associated stabilizing mean. Set

$$q=\frac12(1-p)-2\alpha^2,\qquad p=\pm\sqrt{1+16\alpha^2-16\alpha^4}.$$

Sign conjecture. The inequality

$$L_\alpha-\mathcal{R}(B_p,L_\alpha,B_q)<0$$

holds for $\lvert\alpha\rvert<\frac12$.

This statement concerns the sign needed to determine sub- or super-stabilizability of $L_\alpha$ by power means. The source presents it as motivated by numerical experiments; its resolution is not supplied.