Let $M \subset \mathbb{R}^d$ be a piecewise smooth $2$-manifold. Let $C$ be a polyhedral complex that covers $\mathbb{R}^d$ and contains faces of dimension $[0,d]$. Since $M$ is a $2$-manifold, we can perturb $M$ so that it only intersects faces of $C$ of dimension $d-2$ or greater. Can we smoothen $M$ into a smooth $2$-manifold $\tilde{M}$ such that $d(M,\tilde{M}) \leq \epsilon$ and $\tilde{M}$ intersects every face of $C$ that is intersected by $M$ and no others?