Third-order diagonal rational-function radius conjecture

Let $s\geq 2$, and let $\widehat{R}_{s/s,3}$ denote the maximal radius of absolute monotonicity in the restricted rational-function class used by the paper. Third-order diagonal radius conjecture.

$$\widehat{R}_{s/s,3}=s-1+\sqrt{s^2-1}.$$

The source presents this as stronger than an earlier conjecture, but the supplied text gives no resolution; it therefore remains open.