Lyapunov diagonal stability conjecture for global stability of linear-threshold networks

Let $D\in\mathbb{D}^n_+$ be a diagonal dissipation matrix, $W\in\mathbb{R}^{n\times n}$ a synaptic matrix, and $u\in\mathbb{R}^n$. Consider the linear-threshold network

$$\dot x=-Dx+[Wx+u]_0^1.$$

Write $\mathcal{LDS}$ for the class of Lyapunov diagonally stable matrices, and call an equilibrium globally asymptotically stable if every trajectory converges to it asymptotically.

LDS implies GAS of LTN equilibria. If $W-D\in\mathcal{LDS}$, then for every $u\in\mathbb{R}^n$, the linear-threshold network admits a unique globally asymptotically stable equilibrium.

The conjecture isolates Lyapunov diagonal stability of the effective interaction matrix $W-D$ as a structural sufficient condition for global asymptotic stability. It is presented as a conjecture in the source, and no resolution is supplied in the provided text.