Let $n\in\mathbb{N}$. For $0\le l\le n$ consider \begin{equation} b_l:=4^{-l} \sum_{j=0}^l \frac{\binom{2 l}{2 j} \binom{n}{j}^2}{\binom{2 n}{2 j}}\text{.} \end{equation} Do you know a technique how to prove that \begin{equation} b_l\ge b_n\text{,$\quad 0\le l\le n-1$?} \end{equation} Going through a long list of binomial identities I did not find epiphany.
Addition: Plot of $b_l$ for $n=20$. $b_l$ for $n=20$">