Vortex lines affect the spatial order-parameter distribution in superfluid ³He–B owing to superflow circulating around vortex cores and due to the interaction of the order parameter in the core and in the bulk as a result of superfluid coherence over the whole volume. The step-like change of the latter contribution at 0.6T _c (at a pressure of 29 bar) signifies the transition from axisymmetric cores at higher temperatures to broken-symmetry cores at lower temperatures. We extended earlier measurements of the core contribution to temperatures below 0.2T _c, in particular searching for a possible new core transition to lower symmetries. As a measuring tool we track the energy levels of magnon condensate states in a trap formed by the order-parameter texture.