Semistability of semi-log canonical hypersurface families

Consider the two-dimensional semi-log canonical hypersurface singularities in the families labelled (1)--(3) in the paper. Semistability conjecture. All singularities in these three families are semistable. The authors explain that their degeneration technique does not handle simple elliptic singularities and identify this conjecture as the remaining statement needed for the claimed classification; they also relate it to Mumford's discussion.