Rump's 100 Euro Conjecture for matrices

Let $A\in\mathbb{R}^{n\times n}$, let $e=(1,\dots,1)^T\in\mathbb{R}^n$ be the all-ones vector, and write $|A|$ for the entrywise absolute value of $A$. Suppose that

$$|A|e=ne.$$

Rump's 100 Euro Conjecture. There exists $x\in\mathbb{R}^n\setminus\{0\}$ such that

$$|Ax|\ge |x|.$$

The condition means that every row of $A$ lies on the boundary of the $\ell_1$-ball of radius $n$. The paper's abstract says that this conjecture is confirmed using a finite-dimensional reformulation of Ball's plank theorem, so its status is solved.