Smooth monomial Togliatti systems of cubics

The goal of this paper is to solve the conjecture stated in a paper of Mezzetti, Mir\'o-Roig, Ottaviani and classify all smooth minimal monomial Togliatti systems of cubics. More precisely, we classify all minimal monomial artinian ideals generated by cubics, failing the weak Lefschetz property and whose apolar cubic system defines a smooth toric variety or, equivalently, we classify all minimal monomial artinian ideals generated by cubics whose apolar cubic system defines a smooth toric variety satisfying at least a Laplace equation of order 2.