Assume $u, v \in \mathcal{D}'(\mathbb{R}^n)$ are distributions with compact support. Denote by $\operatorname{WF}(\bullet) \subset T^*\mathbb{R}^n \setminus 0$ the wavefront set of a distribution $\bullet$. If $\operatorname{WF}(u) \cap \operatorname{WF}(v) = \emptyset$, 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, $\operatorname{supp}(u)^c \cup \operatorname{supp}(v)^c = \mathbb{R}^n$?

For the background on the wavefront set and distribution theory, see Chapter 8 of Hörmander's book The Analysis of Linear Partial Differential Operators I. For products under the wavefront set condition, see Theorem 8.2.10 of the same book. I've also asked this question on Mathematics StackExchange.