In two dimensions, the non-unitary class of conformal minimal models, M(2,2m+1), has been recently conjectured to arise as renormalization-group fixed points of scalar field theories with complex iϕ²ᵐ⁻¹ interaction, m∈ N, m≥2. We test a variation of this conjecture through the perturbative study of two separate long-range constructions based on respectively the minimal model and its potential Landau-Ginzburg formalism. For m>2, inconsistencies are found when subsequently relating both constructions. In contrast, the long-range Lee-Yang model, the m=2 case, is shown to be analogue to the long-range Ising model.