The Apéry-sequence Brun criterion conjecture

Let $A_n$ and $B_n$ be the Apéry numbers, let $e_n=2\cdot(\operatorname{lcm}\{1,2,\ldots,n\})^3$, and set

$$x_n=e_nA_n,\qquad y_n=e_nB_n.$$

Let $\Delta$ be the forward difference operator, $\Delta x_n=x_{n+1}-x_n$, and define

$$\delta_n=\Delta\left(\frac{\Delta y_n}{\Delta x_n}\right).$$

Apéry-sequence Brun criterion conjecture. There is an unbounded subsequence of positive integers $n_k\to\infty$ such that $\delta_{n_k}<0$. The sequence $y_n/x_n$ is increasing and converges to $\zeta(3)$; hence, if the conjecture holds, Brun's criterion would imply that $\zeta(3)$ is irrational. The source supplies no proof or resolution.