Ill-posedness of the pure-noise Dean-Kawasaki equation

We prove that the Dean-Kawasaki-type stochastic partial differential equation $$\mathrm{\partial }\rho =\mathrm{\nabla }\cdot (\sqrt{\rho }\xi )+\mathrm{\nabla }\cdot (\rho H(\rho ))$$ with vector-valued space-time white noise $\xi$, does not admit solutions for any initial measure and any vector-valued bounded measurable function $H$ on the space of measures. This applies in particular to the pure-noise Dean-Kawasaki equation ($H\equiv 0$). The result is sharp, in the sense that solutions are known to exist for some unbounded $H$.