The four-bar linkage pop-reachability conjecture

Let a four-bar linkage consist of four bars with fixed lengths, and consider configurations of the linkage with one bar fixed. Two mobile vertices are adjacent when they are consecutive along the linkage, and a pop is the operation that reflects one mobile part across the line through its neighboring vertices. A configuration is reachable by smooth motion if it lies in the same continuous-motion component as the initial configuration, and the neighborhood of a configuration is the set of configurations that can be reached arbitrarily closely.

Four-bar linkage pop-reachability conjecture. For almost every four-bar linkage, the neighborhood of any configuration that can be reached by smooth motion can also be reached by a sequence of alternating pops of two adjacent vertices.

This conjecture asserts dense pop-reachability of every configuration in the smooth-motion reachable set, even when one bar is fixed and pops are restricted to two mobile vertices. The paper proves the claim under additional assumptions on the bar lengths; the general assertion remains open.