The spherical representation conjecture for smooth valuations

Let $i$ be a nonnegative integer, let $\mathbb{R}^{i+1}$ be Euclidean space, and let $\operatorname{Val}_i^{\infty}(\mathbb{R}^{i+1})$ denote the smooth valuations of degree $i$ on it. For a convex body $K$ in $\mathbb{R}^{i+1}$, write $S(K)$ for its surface-area measure on the sphere $\mathbb{S}^i$. Spherical representation conjecture. If $\varphi$ is a smooth valuation of degree $i$ on $\mathbb{R}^{i+1}$, then there exists a smooth function $f$ on the sphere $\mathbb{S}^i$ such that

$$\varphi(K)=\int_{\mathbb{S}^i} f\,dS(K)$$

for every convex body $K$ in $\mathbb{R}^{i+1}$. This would give the representation needed for the asserted inclusion of smooth odd valuations in the relevant subspace; the parser supplies no evidence that the fact is resolved, so its status remains open.