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

Smooth monomial Togliatti systems of cubics