The SO(8) invariant interaction piece in Fidkowski and Kitaev's model

Question

In the paper "The effects of interactions on the topological classification of free fermion systems", the authors demonstrate the existence of a quartic interaction W involving the 8 majorana operators $c_1 \ldots c_8$ (eq. 8) which is invariant under an SO(7) symmetry that can produce a unique ground state invariant under SO(7) as well as time-reversal. They also demonstrate that an SO(8) invariant interaction, which they call V does not work. I want to know the precise form of V which is not explicitly stated in their paper. The only thing the authors say is that V is also quartic in the fermion operators (above eq. 15) and is related to the quadratic Casimir.

I am confused because the 16 dimensional Hilbert space transforms as the spinor representation of SO(8) and is reducible into the two chiral irreps ($8_+$ and $8_-$). From Schur's lemma, the only SO(8) invariant term that can be written is proportional to the projector onto the two 8 dimensional irreps. Since the two irreps are distinguished by the eigenvalue of the chirality operator ($\Gamma_c \sim c_1 c_2 \ldots c_8$), I would think that the term V is proportional to $\Gamma_c$. But this is clearly not quartic.