Conjecture on the sign of the $M_4$ and power-mean difference

Let $M_4$ be the mean defined in the paper, let $B_p$ and $B_q$ be power means, and let $\mathcal{R}(B_p,M_4,B_q)$ denote their associated stabilizing mean. Set

$$p=3-2q,\qquad q=\frac12(3\pm\sqrt{21}).$$

Sign conjecture. Then

$$M_4-\mathcal{R}(B_p,M_4,B_q)>0.$$

The claim records the positive sign obtained from the asymptotic expansion relevant to stabilizability of $M_4$ by power means. The source provides no resolution evidence, so the conjecture remains open.