Is there supersymmetric Lee-Yang fixed point in three dimensions?

The supersymmetric Lee-Yang model is arguably the simplest interacting supersymmetric field theory in two dimensions, albeit non-unitary. A natural question is if there is an analogue of supersymmetric Lee-Yang fixed point in higher dimensions. The absence of any $mathbb{Z}_2$ symmetry (except for fermion numbers) makes it impossible to approach it by using perturbative $epsilon$ expansions. We find that the truncated conformal bootstrap suggests that candidate fixed points obtained by the dimensional continuation from two dimensions annihilate below three dimensions, implying that there is no supersymmetric Lee-Yang fixed point in three dimensions. We conjecture that the corresponding phase transition, if any, will be the first order transition.