The coloured-cycle symmetry conjecture for inverse linear forms

Let $G$ be a coloured $n$-cycle, and let $I(\mathcal{L}^{-1})$ denote the ideal associated with the inverse linear concentration model. A binomial linear form is a linear polynomial involving two monomials, and it is induced by symmetries when it arises from the action of graph symmetries on the model.

Coloured-cycle symmetry conjecture. All binomial linear forms in $I(\mathcal{L}^{-1})$ are induced by symmetries.

The conjecture asserts that, for coloured cycles, graph symmetries account for every binomial linear relation in the inverse model. The statement is presented as a conjecture after computations for the $3$-, $4$-, and $5$-cycles; no resolution is supplied here.