Non-conservatism of the graph hierarchy for robust feedback stabilization

Let $M\in\mathbb{Z}^{+}$ and let

$$\mathcal{F}=\{(A_i,B_i)\in\mathbb{R}^{n\times n}\times\mathbb{R}^{n\times m}\mid i\in\{1,\ldots,M\}\}.$$

Suppose that the switched system under consideration is robust feedback stabilizable. Non-conservatism conjecture. There exists an $l\in\mathbb{N}$, sufficiently large, such that the conditions in the stated corollary based on the graph $\mathcal{H}^{l}(M)$ are feasible. The claim asks whether increasing the graph order eventually removes the conservatism of the sufficient conditions for robust feedback stabilization; the corresponding hierarchy is known to be non-conservative for the stability problem, whereas this robust stabilization question remains open.