Clique convergence conjecture for locally cyclic triangulations of a disc

Let $G$ be a locally cyclic graph with boundary and minimum degree $\delta\ge 4$ triangulating a disc. Such a graph is clique null if its clique dynamics converges to the one-vertex graph. Disc convergence conjecture. The graph $G$ is clique convergent; in fact, it is clique null. This conjecture concerns the exceptional compact surface with boundary that is not covered by the known existence of clique-divergent triangulations, and the paper presents it as an open conjecture.