Chen–Xia conjecture on infinite log-convexity of the Apéry sequences

The Apéry numbers are the sequences

$$A_n=\sum_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2,\qquad B_n=\sum_{k=0}^n\binom{n}{k}^2\binom{n+k}{k},\qquad n=0,1,2,\ldots.$$

A sequence is infinitely log-convex when every sequence obtained by iterating the log-convexity operator remains log-convex. Chen–Xia's conjecture. Both $(A_n)_{n\ge 0}$ and $(B_n)_{n\ge 0}$ are infinitely log-convex sequences. Chen and Xia proposed this conjecture for the Apéry numbers, which arose in Apéry's proof of the irrationality of $\zeta(3)$. Its status is not resolved in the supplied source.