The Catalan–Larcombe–French sequence is infinitely log-convex

Let $\{P_n\}_{n\geq 0}$ be the Catalan–Larcombe–French sequence. A sequence is infinitely log-convex if every sequence obtained by repeatedly applying the operator $\mathcal{L}(a)_n=a_n^2-a_{n-1}a_{n+1}$ is log-convex wherever defined. Infinite log-convexity conjecture. The Catalan–Larcombe–French sequence is $\infty$-log-convex. The paper has established strict 2-log-convexity, and the conjecture proposes that this property continues under arbitrarily many iterations.