An inequality for combination sums

Question

I encountered this problem in my research. I want to prove that \begin{equation}\begin{split} f(r)&=\frac {1}{r} \ln \sum_{p=0}^{r} \sum_{q=0}^{r-p} \frac{C_{2r}^{2p}C_{2r-2p}^{2q}}{(2p+1)(2q+1)(2r-2p-2q+1)}\\ &=\frac {1}{r} \ln \sum_{p=0}^{r} \sum_{q=0}^{r-p} \frac{(2r)!}{(2p+1)!(2q+1)!(2r-2p-2q+1)!} \end{split}\end{equation} is an increasing function of positive integer $r$. Furthermore, does $$\lim_{r\to\infty}{f(r)}$$ exist? I guess that the expression of $f(r)$ can be simplified by combinatorial techniques, but I don't know how to simplify it. The curve of $f(r)$ is shown below: