Attractivity of a system with state-dependent transitions

Let $$A\in\mathbb{R}^{n\times n}$$ and consider the following dynamical system: $$\frac{\mathrm{d}x(t)}{\mathrm{d}t} = -x(t)+\max\{0,Ax(t)\}, \ \ \ \ x(0)\in\mathbb{R}^n,$$ where $$\max\{\cdot\}$$ acts element-wise on vectors.

True or false. If all possible diagonal blocks of $$-I+A$$ (including $$-I+A$$ itself and the diagonal entries of $$-I+A$$) have eigenvalues with (strictly) negative real part, then $$x(t)\to 0$$ as $$t\to\infty$$, for all $$x(0)$$.

A large amount of numerical simulations support the above claim. However a formal proof still remains elusive to me.