We provide sufficient normal curvature conditions on the boundary of a domain D \subset \mathbb{R}^4 to guarantee unboundedness of the bilinear Fourier multiplier operator \mathrm{T}_D with symbol \chi_D outside the local L^2 setting, i.e., from L^{p_1} ( \mathbb{R}^2) \times L^{p_2} ( \mathbb{R}^2) \rightarrow L^{p_3'} ( \mathbb{R}^2) with \sum \frac{1}{p_j} = 1 and p_j <2 for some j . In particular, these curvature conditions are satisfied by any domain D that is locally strictly convex at a single boundary point.