Density conjecture for smooth curvature measures

Let $n$ be a positive integer, let $\operatorname{Curv}_k(\mathbb{R}^n)$ be the degree-$k$ space of curvature measures with its topology induced by compact convergence, and let $\operatorname{Curv}_k^{sm}$ denote the subspace of smooth curvature measures. Density conjecture. For $k\in\{0,\ldots,n-2\}$, the subspace $\operatorname{Curv}_k^{sm}$ is dense in $\operatorname{Curv}_k(\mathbb{R}^n)$. The paper confirms this conjecture in degrees $0$ and $n-2$; the statement is also straightforward in degrees $n-1$ and $n$, but remains open for the other degrees covered by the conjecture.