Haemers' minimum energy conjecture for Seidel matrices

Let $A$ be a symmetric matrix of order $n$ whose diagonal entries are zero and whose off-diagonal entries belong to $\\{\pm 1\\}$. Let $J_n$ be the all-ones matrix and $I_n$ the identity matrix, both of order $n$. Haemers' minimum energy conjecture. The minimum value of the trace norm is

$$\min \Vert A \Vert_1=\Vert J_n-I_n\Vert_1=2n-2.$$

This conjecture concerns the minimum energy of Seidel matrices of simple graphs and is stated here as the motivation for the paper. The supplied context indicates that Akbari et al. proved it, so its status is solved.