Ratio log-concavity conjecture for Sun's R-sequence

Let

$$R_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}\frac{1}{2k-1},\qquad n=0,1,2,\ldots,$$

and let $r_n=R_n/R_{n-1}$ for $n\geq 1$. A sequence is log-concave when its terms satisfy $a_n^2\geq a_{n-1}a_{n+1}$ wherever defined.

Ratio log-concavity conjecture. The sequence $\{r_n\}_{n\geq 4}$ is log-concave, equivalently, $R_n$ is ratio log-concave for $n\geq 4$.

This conjecture is used to establish the decreasing behavior of the root-ratio sequence in the preceding conjecture, and the surrounding proof indicates that it is resolved in the paper.