The augmented operator of a surjective partial differential operator with constant coefficients need not be surjective

For $d\ge 3$ we give an example of a constant coefficient surjective differential operator $P(D):{\mathcal{D}}^{\mathrm{\prime }}(X)\to {\mathcal{D}}^{\mathrm{\prime }}(X)$ over some open subset $X\subset {\mathbb{R}}^d$ such that $P^{+}(D):{\mathcal{D}}^{\mathrm{\prime }}(X\times \mathbb{R})\to {\mathcal{D}}^{\mathrm{\prime }}(X\times \mathbb{R})$ is not surjective, where $P^{+}(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_{d+1}):=P(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_d)$. This answers in the negative a problem posed by Bonet and Doma'nski in \cite[Problem 9.1]{Bonet}.