Alternating log-convexity and log-concavity of iterated Fennessey-Larcombe-French ratios

Let $\mathcal{R}$ act on a sequence $\{S_n\}_{n\geq 0}$ by

$$\mathcal{R}(\{S_n\}_{n\geq 0})=\{S_{n+1}/S_n\}_{n\geq 0}.$$

Let $\{V_n\}_{n\geq 1}$ be the Fennessey-Larcombe-French sequence. Alternating $V_n$ ratio conjecture. For all integer $k\geq 1$, the sequence $\mathcal{R}^k(\{V_n\}_{n\geq 1})$ is log-convex if $k$ is odd and log-concave if $k$ is even. This is a further conjecture on the alternating behavior of iterated ratio sequences associated with the Fennessey-Larcombe-French sequence; no resolution is given in the supplied text.