Il'yuta's monotone-cycle conjecture for Coxeter–Dynkin diagrams

Assume $n\equiv2\pmod 4$. Let $D$ be a Coxeter–Dynkin diagram, with vertices given by a distinguished basis and edge weights $\langle\delta_i,\delta_j\rangle$. A monotone cycle is a sequence of vertices $(\delta_{i_1},\ldots,\delta_{i_k})$ with $i_1<\cdots<i_k$, in which each $\delta_{i_j}$ is connected to $\delta_{i_{j+1}}$, with indices taken cyclically. Let ${\mathcal D}$ denote the collection of such diagrams, and let the modality be the modality of the singularity. Il'yuta's monotone-cycle conjecture. The minimum over all $D\in{\mathcal D}$ of the smallest number of edges that must be deleted so that $D$ contains no monotone cycles is equal to the modality of the singularity. The conjecture was disproved by counterexamples for bimodal singularities.