Profile conjecture for local defect configurations

Consider defects in the Landau–de Gennes model, including $k$-disclination lines whose profiles are obtained from the corresponding profile equations. Profile conjecture. The local configuration of defects can be described by their corresponding profiles. The authors state that disclination lines with the same topological charge are homotopically equivalent and that their local tensor fields can therefore be described by these profiles; no resolution is given.