Alternating log-concavity and log-convexity of iterated ratio sequences

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

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

Alternating ratio-sequence conjecture. For all integer $k\geq 1$, the sequence $\mathcal{R}^k(\{P_n\}_{n\geq 0})$, except for the first $k$ terms at the beginning, is log-concave if $k$ is odd and log-convex if $k$ is even. The source introduces this as a conjecture about iterated ratio transforms and does not provide a proof or resolution.