Let $h_0>0$, let $V\in C^\infty(\mathbb{R}^n;\mathbb{R})$, let $E\in\mathbb{R}$, and put \begin{align*} P_h=-h^2\Delta+V(x)-E. \end{align*} Let \begin{align*} p(x,\xi)=|\xi|^2+V(x)-E \end{align*} be its semiclassical principal symbol. Let $K\subset T^*\mathbb{R}^n$ be compact. Assume there are an open neighbourhood $U$ of $K$ and a constant $c>0$ such that \begin{align*} |p(x,\xi)|\ge c \end{align*} for all $(x,\xi)\in U$, and assume all derivatives of $V$ that occur on the projection of a fixed compact neighbourhood of $U$ are bounded. Let $u_h\in\mathcal{S}'(\mathbb{R}^n)$ be semiclassically tempered: for every compactly supported symbol $a\in C_c^\infty(T^*\mathbb{R}^n)$ and every $s\in\mathbb{R}$ there are constants $C,Q>0$ with \begin{align*} \|\operatorname{Op}_h(a)u_h\|_{H_h^s}\le Ch^{-Q} \end{align*} for $0<h\le h_0$. Suppose that for one cutoff $\theta\in C_c^\infty(U)$ with $\theta=1$ on a neighbourhood of $K$, \begin{align*} \operatorname{Op}_h(\theta)P_hu_h=O(h^\infty) \end{align*} in $H_h^s(\mathbb{R}^n)$ for every $s\in\mathbb{R}$. Then $K\cap\operatorname{WF}_h(u_h)=\varnothing$. Equivalently, for every $\chi\in C_c^\infty(T^*\mathbb{R}^n)$ with $\operatorname{supp}\chi\subset U$ and $\theta=1$ on a neighbourhood of $\operatorname{supp}\chi$, \begin{align*} \operatorname{Op}_h(\chi)u_h=O(h^\infty) \end{align*} in $H_h^s(\mathbb{R}^n)$ for every $s\in\mathbb{R}$.