The crucial observation is that TLSG defines vertex costs (implicitly all-zero) and edge costs, and asks for vertex signs, whereas TRIANGLE BALANCE defines edge costs and triangle costs, and asks for edge signs.
vertices (edges) in TLSG correspond to edges (triangles) in TRIANGLE BALANCE, and our proposed reduction transforms an original TLSG instance into a TRIANGLE BALANCE instance in which each edge corresponds to exactly one original vertex, and each triangle to exactly one original edge.