  We prove the existence of a positive semidefinite matrix $A \in \mathbb{R}^{n
\times n}$ such that any decomposition into rank-1 matrices has to have factors
with a large $\ell^1-$norm, more precisely $$ \sum_{k} x_k x_k^*=A \quad
\implies \quad \sum_k \|x_k\|^2_{1} \geq c \sqrt{n} \|A\|_{1},$$ where $c$ is
independent of $n$. This provides a lower bound for the Balan--Jiang matrix
problem. The construction is probabilistic.
