Given a constant \(\alpha > 0\), a flat cone \(\mathtt{fC}(\alpha)\) in \(\mathbb{R}^3\) is a set of the form \[ \{ (x,y,0) : |y| < \alpha |x| \}. \] A set \(S\) in the first sub-Riemannian Heisenberg group \(\mathbb{H}\) has the flat cone property if there is some \(\alpha > 0\) such that \[ \mathtt{fC}(\alpha) \cap p^{-1} S = \emptyset \] for all \(p \in \mathbb{H}\). Intrinsic Lipschitz graphs may be defined as those sets in \(\mathbb{H}\) such that, for some open cone \(C \subset \mathbb{H}\) that intersects the \(xy\)-plane nontrivially and satisfies \(C = -C\), we have \[ C \cap p^{-1} S = \emptyset \] for all \(p \in \mathbb{H}\). Since any such open cone contains a flat cone as a subset, it follows that every intrinsic Lipschitz graph in \(\mathbb{H}\) satisfies the flat cone property. In the paper under review, the authors prove the reverse implication for topological surfaces. That is, any topological surface satisfying the flat cone property must locally be an intrinsic Lipschitz graph. This is obtained by noticing the following crucial fact: translating the \(xy\)-plane by a point \((x,y,0)\) results in an affine plane, and the family of planes created by fixing \(y\) and allowing \(x\) to be arbitrary all contain a common line in the \(xz\)-plane.