Genus three Ceresa cycles and limit of archimedean heights

For a one-parameter variation of biextension mixed Hodge structures, Brosnan and Pearlstein showed that the limit of the asymptotic height of the variation is given by a certain limit height of the nilpotent orbit. This limit height depends on the choice of a parameter. In the case of a variation of geometric origin related to Ceresa cycles associated with curves of genus three, after fixing a parameter, we show that this limit height is given by the Deligne splitting of a biextension mixed Hodge structure associated with cycles in the boundary.