Let $\mathcal{D}'_+:=\{T\in \mathcal{D}'(\mathbb{R}): \textrm{supp}(T)\subset [0,\infty)\}$. Here $\mathcal{D}'(\mathbb{R})$ is the usual space of distributions on $\mathbb{R}$, equipped with the weak$\ast$-topology induced by $\mathcal{D}(\mathbb{R})$, and $\mathcal{D}_+'$ is given the subspace topology induced from $\mathcal{D}'(\mathbb{R})$.

Question: Is convolution $\ast:\mathcal{D}'_+ \times \mathcal{D}'_+\rightarrow \mathcal{D}'_+$ separately continuous?