Friedman's $\mathsf{W}\mathsf{D}$ is not parameter-free sequential

Harvey Friedman’s $\mathsf{W}\mathsf{D}$ is a weak set theory given by the following non-logical axioms: $(\mathsf{W})\mathrm{\forall }xy\mathrm{\exists }z\mathrm{\forall }u[u\in z\leftrightarrow (u\in x\vee u=y)]$; $(\mathsf{D})\mathrm{\forall }xy\mathrm{\exists }z\mathrm{\forall }u[u\in z\leftrightarrow (u\in x\wedge u\mathrm{\ne }y)]$. We answer a question raised by Albert Visser which asks whether $\mathsf{W}\mathsf{D}$ is parameter-free sequential. Let $\mathsf{W}\mathsf{D}+\mathsf{E}\mathsf{X}\mathsf{T}$ denote the theory we obtain by extending $\mathsf{W}\mathsf{D}$ with the axiom of extensionality. We show that $\mathsf{W}\mathsf{D}+\mathsf{E}\mathsf{X}\mathsf{T}$, and hence also $\mathsf{W}\mathsf{D}$, is not parameter-free sequential by using forcing to construct a model ${\mathcal{V}}^{\star }$ of $\mathsf{W}\mathsf{D}+\mathsf{E}\mathsf{X}\mathsf{T}$ where $({\mathcal{V}}^{\star },a)\simeq ({\mathcal{V}}^{\star },b)$ for any two elements $a,b$ of ${\mathcal{V}}^{\star }$.