 if and only if there exist (i) an $\mathcal{A}$-ample set $\Pi$ of places for $\varphi$ such that $|\Pi| \leq |\mathcal{A}| + |\mathsf{V}_{0}|$, for some promising set $\mathcal{A}$ of atoms in $\varphi$, and (ii) a map $x \mapsto \pi^{x}$ from $\mathsf{V}_{0}$ into $\Pi$, such that $\pi^{x}$ is the sole place in $\Pi$ at the variable $x$, for $x \in \mathsf{V}_{0}$. 