A counterexample to an endpoint bilinear Strichartz inequality

The endpoint Strichartz estimate $\| e^{it\Delta} f \|_{L^2_t L^\infty_x(\R \times \R^2)} \lesssim \|f\|_{L^2_x(\R^2)}$ is known to be false by the work of Montgomery-Smith, despite being only ``logarithmically far'' from being true in some sense. In this short note we show that (in sharp constrast to the $L^p_{t,x}$ Strichartz estimates) the situation is not improved by passing to a bilinear setting; more precisely, if $P, P'$ are non-trivial smooth Fourier cutoff multipliers then we show that the bilinear estimate $$\| (e^{it\Delta} P f) (e^{it\Delta} P' g) \|_{L^2_t L^\infty_x(\R \times \R^2)} \lesssim \|f\|_{L^2_x(\R^2)} \|g\|_{L^2_x(\R^2)} $$ fails even when $P$, $P'$ have widely separated supports.