A chord diagram is a trivalent graph that looks like a circle with lines going across it called chords. If we make a graph where each vertex represents a chord and there is an edge between vertices if the chords cross eachother inside the circle, this is called the intersection graph. The Intersection Graph Conjecture asks whether or not two chord diagrams with the same intersection graph are equal in the algebra of chord diagrams. A square TLV is a chord diagram such that the intersection graph consists of two 4-cycles that share at most a vertex. I will go over transformations of such diagrams that preserve the intersection graph and make the conjecture true.