Higher-order boundedness conjecture for contractive hBCS matrices

Let $M$ be the matrix associated with a one-dimensional hyperbolic boundary control system, let $\mathbf{U}(s)$ be the corresponding matrix-valued function, let $n$ be the system dimension, and let $\|\cdot\|_{\mathcal{M}}$ denote the relevant matrix-measure norm. Higher-order boundedness conjecture. If

$$\|M\|_{\ell^2\rightarrow\ell^2}<1,$$

then there exists $k_0$ such that

$$\sum_r\left\|\left(\left(M\mathbf{U}(s)\right)^{k_0}\right)_{p,r}\right\|_{\mathcal{M}}<1,\qquad 1\leq p\leq n.$$

This conjecture would imply the positive boundary dissipation conjecture for the diagonalised subclass with $P_0^D=0$. The authors have not found a counterexample, whereas the corresponding non-strict condition can fail, so the conjecture remains open.