Fundamentality conjecture for connected twistless large-type Artin graphs

Let $A_\Gamma$ be a large-type Artin group with defining graph $\Gamma$. The graph $\Gamma$ is fundamental when every subgraph of its intersection graph $I_\Gamma$ that is isomorphic to the barycentric subdivision of $\Gamma$ corresponds to a unique fundamental domain in the Deligne complex $D_\Gamma$.

Fundamentality conjecture. If $\Gamma$ is connected and twistless, then $\Gamma$ is fundamental.

Fundamentality is the combinatorial condition used in the paper to extend rigidity, classification and related results from groups with twistless hierarchies to broader classes. The paper proves fundamentality for groups admitting a twistless hierarchy terminating in twistless stars and for connected twistless graphs whose labels are at least $6$, while the full connected twistless case remains conjectural.