Kindly help me to prove/disprove the following statement.

Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ (where $k$ is a fixed positive integer). Similarly,

Let $B$ be a symmetric matrix of order $n \times n$ with every diagonal entry equal to $0$, and each non-diagonal entry equal to some $\ell$ with $1\leq \ell \leq k<n$.

**For example**:$ A= \begin{pmatrix} \begin{array}{cccc}
0 & 3 & 3 & 3 \\
3 & 0 & 3 & 3 \\
3 & 3 & 0 & 3 \\
3 & 3 & 3 & 0 \\
\end{array}\end{pmatrix}$,
$B=\begin{pmatrix}
\begin{array}{cccc}
0 & 3 & 1 & 2 \\
3 & 0 & 2 & 3 \\
1 & 2 & 0 & 1 \\
2 & 3 & 1 & 0 \\
\end{array}
\end{pmatrix}$.

Prove/disprove the following statement:

**Suppose $\lambda_1, \lambda_2, ..., \lambda_n$ are eigenvalues of $A$ and $\mu_1, \mu_2, ..., \mu_n$ are eigenvalues of $B$. Then $$\sum_{i=1}^{n}{|\lambda_i|}\geq \sum_{i=1}^{n}{|\mu_i|}.$$**

Note that $A$ and $B$ are the distance matrices of some vertices(diameteral) of $G_1$ and $G_2$ respectively.