 Solution

The inclusion $\mathcal{C}\subseteq \mathcal{D}$ is a structure embedding between first-order models of the same complete theory, but it is not an elementary embedding, as the existential partner first-order formula at $u$ shows. Therefore the first-order theory is not model-complete. The facts $\mathcal{B}⪯\mathcal{C}$ and $\mathcal{B}⪯\mathcal{D}$ do not make the intervening inclusion elementary.