Sun's ratio monotonicity conjecture for the sequence S_n

Let

$$S_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}(2k+1),\qquad n=0,1,2,\ldots.$$

Sun's ratio monotonicity conjecture. The sequence $\{S_{n+1}/S_n\}_{n\geq 3}$ is strictly increasing to the limit $9$, and the sequence $\{\sqrt[n+1]{S_{n+1}}/\sqrt[n]{S_n}\}_{n\geq 1}$ is strictly decreasing to the limit $1$. The paper's abstract states that it gives an affirmative answer to this conjecture, so the claim is solved in the source paper.