The Halanay–Smith criterion for strong stability with distributed delays

Let $M\in\mathcal G$, and let $\Sigma_M$ denote the associated linear functional system. Write

$$\rho_{\mathrm{HS}}(M):=\sup_{\xi\in\mathfrak C}\rho\left(\int_{-1}^0e^{i\xi(\theta)}\,dM(\theta)\right),$$

where $\mathfrak C=\{\xi\colon[-1,0]\to\mathbb R:\xi\text{ is Borel-measurable}\}$. A system is strongly stable if every admissible pushforward $\varphi_*\mu$ is exponentially stable, and locally strongly stable if this holds for all admissible pushforwards sufficiently close to the identity.

Halanay–Smith criterion. Let $M\in\mathcal G$. The following statements are equivalent:

1. $\rho_{\mathrm{HS}}(M)<1$.
2. The system $\Sigma_M$ is strongly stable.
3. The system $\Sigma_M$ is locally strongly stable.

This conjecture proposes that the spectral-radius condition extends the Halanay–Smith stability criterion from the previously established setting to general distributed-delay systems. The supplied text does not state whether the equivalence has been proved or refuted.