Six-vertex criterion for infinite-dimensional Fomin–Kirillov algebras

Let $G$ be a graph on six vertices, and let $\mathcal E_G$ be its associated algebra. The seven graphs displayed in Figure~ are the specified six-vertex graphs.

Six-vertex infinite-dimensionality conjecture. The algebra $\mathcal E_G$ is infinite-dimensional if and only if $G$ contains a subgraph isomorphic to one of the graphs shown in Figure~.

The conjecture is motivated by computations: the algebras are finite-dimensional for all graphs on at most five vertices, while some six-vertex examples appear infinite-dimensional. The authors state that they cannot yet prove infinite-dimensionality for any example; in particular, the conjecture would imply that $\mathcal E_6$ is infinite-dimensional.