The triple-diagram relations-between-relations conjecture

A triple diagram is a diagram in the paper's class, and a minimal triple diagram is one with no reducible features under the allowed simplification moves. Fix a connectivity of diagrams and let $C$ be the $2$-complex whose vertices are minimal triple diagrams and whose edges are $2\leftrightarrow2$ moves. Its $2$-cells are squares for two different $2\leftrightarrow2$ moves in non-interfering parts of the diagram, pentagons as in the left side of Figure 1 or with reversed orientation, and decagons as in the right side of Figure 1.

Relations-between-relations conjecture. The $2$-complex $C$ has precisely these types of $2$-cells.

This conjecture describes relations among the local moves relating minimal triple diagrams. The surrounding discussion identifies the pentagonal and decagonal cells geometrically; the supplied text does not state whether the conjecture has been proved or disproved.