Let $a\in S(\langle \xi\rangle^2)$ be real-valued for the standard semiclassical Euclidean symbol class, with seminorms measured by the usual estimates \begin{align*} |\partial_x^\alpha\partial_\xi^\beta a(x,\xi)|\le C_{\alpha\beta}\langle \xi\rangle^{2-|\beta|}. \end{align*} Assume $a(x,\xi)\ge 0$ for all $(x,\xi)\in T^*\mathbb R^n$. Then there exist constants $C>0$ and $h_0>0$, depending on finitely many of these symbol seminorms of $a$, such that \begin{align*} (\operatorname{Op}_h^w(a)u,u)_{L^2}\ge -Ch^2\|u\|_{L^2}^2 \end{align*} for every $u\in \mathcal S(\mathbb R^n)$ and $0<h\le h_0$.