Monotonicity conjecture for the convergence-rate quotients of the canonical iterates

The sequence $(\kappa_n)_{n\in\mathbb N_0}$ provides approximations to the stribolic constant $\kappa$. Define the convergence-rate quotients by

$$\vartheta_n:=\frac{\kappa_n-\kappa_{n+1}}{\kappa_{n-1}-\kappa_n},\qquad n\in\mathbb N.$$

Monotonicity conjecture. For all $m\in\mathbb N$, one has

$$\vartheta_{2m}<\vartheta_{2m+2}<\vartheta_{2m+3}<\vartheta_{2m+1}.$$

The inequalities are suggested by the computed values for $2\leq n\leq22$ and would provide sharper estimates for $\kappa$ through iterated extrapolation.