Licci Level Stanley-Reisner Ideals with Height Three and with Type Two

Via computer-aided classification we show that the following three conditions are equivalent for level* squarefree monomial ideals I with codimension 3, with Cohen-Macaulay type 2 and with \( \dim S/I \le 4\): (1) \(IS_{{\mathfrak m}}\) is licci, (2) the twisted conormal module of I is Cohen-Macaulay, (3) \(S/I^{(2)}\) is Cohen-Macaulay, where S is a polynomial ring over a field of characteristic 0 and \({\mathfrak m}\) is its graded maximal ideal.