Infinite log-convexity conjecture for the Baxter numbers

Let $\{B_n\}_{n\geq 1}$ be the sequence of Baxter numbers. The Baxter-number infinite log-convexity conjecture. The sequence

$$\{B_n\}_{n\geq 1}$$

is infinitely log-convex. The paper proves eventual $3$-log-convexity from index $882712$ onward, but does not prove $3$-log-convexity from the beginning; the full infinite log-convexity claim remains open.