Stabilizing solution conjecture for the line-graph structural space

Let $H$ be the stabilizing positive semidefinite solution of the reduced discrete algebraic Riccati equation under Assumption~, and let ${\mathcal V}_{\mathcal L}$ denote the structural subspace associated with the line graph. Stabilizing solution conjecture. Under Assumption~, the matrix $H$ belongs to ${\mathcal V}_{\mathcal L}$:

$$H\in{\mathcal V}_{\mathcal L}.$$

Exhaustive searches over all directed trees with at most six vertices found this property in every tested case. The conjecture proposes that the observed refinement ${\mathcal V}_{\mathcal L}$ contains the stabilizing solution generally, although the paper proves containment only in the weaker space ${\mathcal V}_{\mathcal B}$ and gives a counterexample showing that the corresponding invariance proposition cannot be extended to ${\mathcal V}_{\mathcal L}$.