Overlap between switching methods from different orthogonal matrices

Prove or disprove that distinct indecomposable orthogonal matrices $Q_1$ and $Q_2$ of the same size cannot induce switching methods with the same $Q$-respecting vectors up to a permutation that also maps the associated switching graphs $\Gamma_1$ and $\Gamma_2$ to one another.

Background

The paper’s counting framework can combine asymptotic estimates for switching methods associated with the same orthogonal matrix when their switching graphs are sufficiently distinct. For different orthogonal matrices, substantial overlap may occur when their admissible attachment vectors and switching graphs correspond under a permutation.

The authors report that they know of no such pair of matrices, but they cannot rule out its existence. Establishing nonexistence would justify broader aggregation of the asymptotic bounds across different switching methods; constructing an example would demonstrate an obstruction to such aggregation.