I came across the following while doing some related proof; It seems easy to prove $\quad$

$1$) An $n\times n$ matrix $U$ that is unitary has up to permuting columns a diagonal such the modulus of each entry $\le \dfrac{\sqrt{2}}{2}$, $\quad$

And more difficult:

$2$) There are no $n\times n$ unitaries with diagonal $D$ where $|d_{i,i}|>\dfrac{1}{\sqrt{n-1}}$ for all $i$, $i$, unless $U$ is a direct sum of unitaries in ${\mathbb{M}}_k$, $k<n$. I am searching for a proof or related facts Thanks.