  Recently, Grabowska and Kaplan constructed a four-dimensional lattice
formulation of chiral gauge theories on the basis of the chiral overlap
operator. At least in the tree-level approximation, the left-handed fermion is
coupled only to the original gauge field~$A$, while the right-handed one is
coupled only to the gauge field~$A_\star$, a deformation of~$A$ by the gradient
flow with infinite flow time. In this paper, we study the fermion one-loop
effective action in their formulation. We show that the continuum limit of this
effective action contains local interaction terms between $A$ and~$A_\star$,
even if the anomaly cancellation condition is met. These non-vanishing terms
would lead an undesired perturbative spectrum in the formulation.
