Assume $u, v \in \mathcal{D}'(\mathbb{R}^n)$ are distributions with compact support. Denote by $\mathrm{WF}(\bullet) \subset T^*\mathbb{R}^n \setminus 0$ the wavefront set of a distribution $\bullet$. If $\mathrm{WF}(u) \cap \mathrm{WF}(v) = \emptyset$, then their product $uv$ is well defined. If $uv = 0$, does this imply that at every point one of the two distributions vanish, that is, $\mathrm{supp}(u)^c \cup \mathrm{supp}(v)^c = \mathbb{R}^n$?

For the background on the wavefront set and distribution theory, see Chapter 8 of Hörmander's book. For products under the wavefront set condition, see Theorem 8.2.10 of the same book. I've also asked this question here.