Linear relations of colored Gaussian cycles

A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. Marigliano and Davies conjectured that every linear binomial that appears in the vanishing ideal of an undirected colored cycle corresponds to a graph symmetry. We prove this conjecture for 3,5, and 7 cycles and disprove it for colored cycles of any other length. We construct the counterexamples by proving the fact that the determinant of the concentration matrices of two colored paths can be equal even when they are not identical or reflection of each other. We also explore the potential strengthening of the conjecture and prove a revised version of the conjecture.