TITLE: Is this lower bound for a norm of some complex matrices true?
QUESTION [15 upvotes]: Let $A = [a_{ij}]_{n\times n}$ be a Hermitian matrix, such that $|a_{ij}| =1$ for $i \neq j$, and $a_{ii} = 0$ for each $i$.
I am interested in a tight lower bound of $\|A\|_*:=\sum_{i=1}^n |\lambda_i(A)|$, where $\lambda_i(A)$'s are eigenvalues of $A$.
Note that, 
by minimizing $\sum_{i=1}^n |\lambda_i|$ over two constraints $\sum_{i=1}^n \lambda_i = 0$ and $\sum_{i=1}^n \lambda_i^2= n(n-1)$, one can obtain $\sqrt{2n(n-1)}$ as a lower bound. But it seems that isn't tight.
On the other hand, if $A := J - I$ (all ones matrix minus identity), then $\sum_i |\lambda_i(A)| = 2(n-1)$. 
Is it true that $2(n-1)$ is actually a lower bound (for large enough matrices, say $n \geq 10$) ?
Remarks:

As Alex's answer below, the minimum of trace norm of such matrices may be less than $2(n-1)$, even for arbitrarily large matrices.
But, as a comment of @fedja, the minimum is $(2+o(1))n$ as $n\to\infty$.

Added:

In the particular case, when $a_{ij}=±1$, the lower bound holds. See this answer below, for an overview of the proof.

REPLY [2 votes]: In the special case $a_{ij} = \pm 1$, $2(n-1)$ is a lower bound, for every $n>0$. 
As a comment by T. Tao above, the problem  resembles 
         the sharp Littlewood conjecture on the minimum of
         the $L^{1}$-norm of polynomials (on the unit circle
         in the complex plane) whose absolute values of
         coefficients are equal to $1$. In
         the special class of polynomials with $\pm 1$
         coefficients, Klemes proved the sharp Littlewood
         conjecture (see here).   

The proof of Klemes gives us the following equality, for an $n\times m$
   matrix $A$ with singular values $\sigma_1,\ldots,\sigma_{r}$ and for $ 0 \leq p \leq 2 $:
           \begin{equation*}  \sum \limits_{i=1}^{r} \vert \sigma_{i} \vert^p= C_p \int_{0}^{\infty} \log \left(1+\sum \limits_{k=1}^{r}
 S_{k}(A^*A) t^{k} \right)t^{-\frac p2 -1}dt,
   \end{equation*}  where $S_k(A^*A)$  stand for the sum of the determinat of $k\times k$ principle  submatrices of $A^*A$ and  $C_p$ is a constant depenting on $p$.

When $A$ is Hermitian, singular values are equal to eigenvalues and by obtaining a  "good" lower bound for  $S_{k}(A^2)$, when $A$ is a matrix of the form described in the question, we can establish the lower bound in the special case $a_{ij} = \pm 1$.