Infinite log-convexity of the Catalan-Larcombe-French transform sequence

Let $\{V_n\}_{n\geq 0}$ be the Fennessey-Larcombe-French sequence, and consider the sequence $\{V_n^2-V_{n-1}V_{n+1}\}_{n\geq 2}$. Infinite log-convexity conjecture. The sequence $\{V_n^2-V_{n-1}V_{n+1}\}_{n\geq 2}$ is infinitely log-convex. This conjecture concerns higher-order log-convexity for a sequence related to the Catalan-Larcombe-French and Fennessey-Larcombe-French sequences. The source notes that Stieltjes moment sequences are infinitely log-convex, suggesting an analytic approach, but does not establish the conjecture.