For every $\Pi_1$-sentence $\varphi(X)$ there is a $d \ge 1$ such that $p\text{-WD}_\varphi \le^{\mathrm{fpt}} p\text{-WSat}(d\text{-CNF})$ [FG06, Lemma 6.37]. Together with $WH_D08_WSatInA1$ this gives $\mathrm{W}[1] \subseteq \mathrm{A}[1]$.

Construction. Write the quantifier-free part of $\varphi = \forall x_1 \dots \forall x_r\, \psi$ in conjunctive normal form $\bigwedge_{i \in I} \bigvee_{j \in J} \lambda_{ij}$. Let $U$ be the elements occurring in the structure's word together with the first $|x| + s\cdot k + r$ elements; the remaining elements occur in no relation and are interchangeable, so a witness can be moved into $U$. Take a propositional variable $Y_{\bar a}$ for every $\bar a \in U^s$ ($s$ the arity of $X$), meaning $\bar a \in X$. For every $i \in I$ and $a_1, \dots, a_r \in U$ form the clause of the literals $\lambda_{ij}(a_1, \dots, a_r)$: a literal $(\neg) X\bar y$ becomes $(\neg) Y_{\bar a}$ for the tuple $\bar a$ of values of $\bar y$; a literal without $X$ is evaluated in $\mathcal A$ and dropped if false, and the whole clause is dropped if it is true. The clauses $Y_{\bar a} \vee \neg Y_{\bar a}$ make every variable occur. Then $\mathcal A \models \varphi(S)$ iff $\{Y_{\bar b} : \bar b \in S\}$ satisfies the formula, and $k$ is unchanged.

Complexity. The formula has $O(|U|^{r+s}\cdot|\varphi|)$ literals; since $|U|$ grows with $k$, the reduction runs in time $(|x|\cdot k)^{O(1)}$, fixed-parameter.