  We investigate the fate of the non-supersymmetric Gross-Neveu-Yukawa fixed
point found by Fei et al in $4-\epsilon$ dimensions with a two-component
Majorana fermion continued to two dimensions. Assuming that it is a fermionic
minimal model which possesses a chiral $\mathbb{Z}_2$ symmetry (in addition to
fermion number parity) and just two relevant singlet operators, we can zero in
on four candidates. Assuming further that the least relevant deformation leads
to the supersymmetric Gross-Neveu-Yukawa fixed point (i.e. fermionic
tricritical Ising model), we can rule out two of them by matching the spin
contents of the preserved topological defect lines. The final candidates are
the fermionic $(11,4)$ minimal model if it is non-unitary, and the fermionic
$(E_{6}, A_{10})$ minimal model if it is unitary. If we further use a
constraint from the double braiding relation proposed by one of the authors,
the former scenario is preferable.
