We prove that the Dean-Kawasaki-type stochastic partial differential equation $$\partial ρ= \nabla\cdot (\sqrt{ρ\,}\, ξ) + \nabla\cdot \left(ρ\, H(ρ)\right)$$ with vector-valued space-time white noise $ξ$, 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 $H$.