  Strong evidence for dual superconformal symmetry in $\mathcal{N} = 6$
superconformal Chern-Simons theory has fueled expectations that the AdS/CFT
dual geometry $AdS_4 \times \mathbb{C} P^3$ is self-dual under T-duality. We
revisit the problem to identify commuting bosonic and fermionic isometries in a
systematic fashion and show that fermionic T-duality, a symmetry originally
proposed by Berkovits & Maldacena, inevitably leads to a singularity in the
dilaton transformation. We show that TsT deformations commute with fermionic
T-duality and comment on T-duality in the corresponding sigma model. Our
results rule out self-duality based on fermionic T-duality for $AdS_4 \times
\mathbb{C} P^3$ or its TsT deformations, but leave the door open for new
possibilities.
