Zhao's log-concavity conjecture for the Fennessey-Larcombe-French sequence

Let $\{V_n\}_{n\geq 0}$ be the Fennessey-Larcombe-French sequence, defined by $V_0=1$, $V_1=8$, and

$$n(n+1)^2 V_{n+1}=8n(3n^2+5n+1)V_n-128(n-1)(n+1)^2V_{n-1},\qquad n\geq 1.$$

A sequence $\{a_n\}$ is log-concave when $a_n^2\geq a_{n-1}a_{n+1}$ for every relevant index. Zhao's conjecture. The Fennessey-Larcombe-French sequence $\{V_n\}_{n\geq 1}$ is log-concave; that is,

$$V_n^2\geq V_{n-1}V_{n+1}\qquad(n\geq 2).$$

The conjecture concerns the log-behavior of the sequence arising from the series expansion of the complete elliptic integral of the second kind. The paper records it after proving that $\{nV_n\}_{n\geq 1}$ and $\{V_n/(n-1)!\}_{n\geq 1}$ are log-concave; its resolution is not specified in the supplied text.