Conjecture on the sign of the Stolarsky and power-mean difference at exceptional parameters

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

$$q=\frac12(1-p)+2\alpha^2, \qquad p^2=\frac{1-12\alpha^2+112\alpha^4-64\alpha^6}{1+4\alpha^2}.$$

Sign conjecture. At $\lvert\alpha\rvert=\frac12\sqrt{2\pm\sqrt3}$,

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

The claim arises from asymptotic comparison of $S_\alpha$ with the stabilizing mean generated by power means. The source gives no resolution evidence beyond the preceding asymptotic calculation, so the conjecture remains open.