Strict separation of Robin and Weyl critical parameters

Let $\beta(\gamma,d)$ be the critical Robin parameter from Theorem~, and let $\beta_W(\gamma,d-1)$ be the point where the second term in the two-term Weyl formula changes sign. Strict-separation conjecture. For every $d\geq1$ and $\gamma>0$,

$$\beta(\gamma,d)>\beta_W(\gamma,d-1).$$

The authors state that they can prove this inequality for some pairs but believe it always holds; the general claim remains open.