Let $A\in\mathbb{R}^{n\times n}$ and consider the following system $$\tag{#}\label{eq:1} \dot{x}(t) = -x(t) + A [x(t)]_+,\ \ \ x(0)=x_0\in\mathbb{R}^n $$ where $[x]_+=\max\{0,x\}$ ($\max\{\cdot\}$ is applied element-wise, if $x$ is a vector).
If all the possible diagonal submatrices of $-I+A$ (including $-I+A$ itself) have eigenvalues with (strictly) negative real part, then the origin is a globally asymptotically stable equilibrium point for \eqref{eq:1}, i.e., $\lim_{t\to \infty} x(t)=0$ for all $x_0\in\mathbb{R}^n$.
Is this claim true?
Extensive numerical evidence seems to suggest that this is indeed the case. Any help/suggestion towards a formal proof is very appreciated.