Irreducibility conjecture for zero-total curvature measures

Let $n$ be a positive integer, let $\operatorname{Curv}_k^\pm(\mathbb{R}^n)$ be the even or odd degree-$k$ curvature measures, and define

$$\operatorname{Z}_k^\pm(\mathbb{R}^n)=\{\Phi\in\operatorname{Curv}_k^\pm(\mathbb{R}^n):\Phi(K,\mathbb{R}^n)=0\text{ for every convex body }K\}.$$

This is an invariant closed subspace under the action of $GL(n,\mathbb{R})$. Irreducibility conjecture. For $k\in\{0,\ldots,n-2\}$, the representation of $GL(n,\mathbb{R})$ on $\operatorname{Z}_k^\pm(\mathbb{R}^n)$ is irreducible. The paper states that this conjecture, together with the density conjecture, is confirmed in degrees $0$ and $n-2$; its validity in the remaining degrees is not established here.