Is there tight estimates for the following logarithmic summation ($\gamma,\gamma'\in(0,1)$ and $\mu,\mu'>0$)

$$\log_2\Bigg(\sum_{t=\frac{n^{}}2-n^\gamma\sqrt{\mu\ln n}}^{\frac{n^{}}2+n^\gamma\sqrt{\mu\ln n}}\quad\sum_{\ell=\frac{n^{}}2-n^\gamma\sqrt{\mu\ln n}}^{\frac{n^{}}2+n^\gamma\sqrt{\mu\ln n}}\quad\sum_{k=\frac t2-n^{\gamma'}\sqrt{\mu'\ln n}}^{\frac t2+n^{\gamma'}\sqrt{\mu'\ln n}}\binom{\ell}{k}\binom{n-\ell}{t-k}\Bigg)?$$

I am hoping it might give $n - f(n)$ bound where $f(n)$ is $\omega(\ln n)$ or at least $\Omega(1)$ for diagonal case of $\gamma=\gamma'=\frac12$ at some $\mu,\mu'>0$.

Relevant problem is in Tight estimates for binomial summation (and perhaps an upper bound **possible** might be $$\log_2\Bigg(\underbrace{n^{2\gamma}\mu(\ln n)}_{\substack{\mbox{coming from}\\\mbox{outer two}\\\mbox{summations}}}\binom{n}{n/4}\Bigg)<n H(\frac14+\epsilon)<0.82n$$ at any $\epsilon>0$ since $t/2=n/4$ might be close to the value that yields the bound (together with loose upper bound from Vandermonde's identity for inner sum)).