I came across the following while doing some related proof; It seems easy to prove. $\quad$
We are in ${\mathbb{M}}_n(\mathbb{C})$, $n>1$:

$1$) Given a unitary $n\times n$ matrix $U$, there is some permutation of the columns such that the modulus of each entry on the diagonal of the resulting matrix is $\le \dfrac{\sqrt{2}}{2}$, $\quad$

And more difficult:

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