The converse of the WLP criterion for level monomial ideals

Let $K$ be a field of characteristic zero, let $R=K[x,y,z]$, and let $I_{\alpha,\beta,\gamma,t}$ denote the level monomial almost complete intersection used in the preceding criterion. The criterion gives three cases ensuring failure or presence of the WLP. Converse conjecture. If $R/I_{\alpha,\beta,\gamma,t}$ fails to have the WLP, then one of cases (1), (2), or (3) holds. This converse is intended to complete the characterization of level monomial ideals in three variables that fail to have the WLP; the supplied text does not state its resolution.