Hameister–Rao–Simpson Hilbert-series identity conjecture for uniform matroids

Let $\mathsf{U}_{k,n}$ be the uniform matroid of rank $k$ on $n$ elements, let $\mathrm{A}(\mathsf{U}_{k,n})$ be its Chow ring, and let $h_{\Delta(\widehat{\mathcal{L}}(\mathsf{U}_{i,n}))}(x)$ be the $h$-polynomial of the chain complex of the proper lattice of flats. Uniform-matroid Hilbert-series identity.

$$\operatorname{Hilb}(\mathrm{A}(\mathsf{U}_{k,n}),x)=\sum_{i=1}^{k}(-1)^{k-i}\binom{n-i-1}{k-i}h_{\Delta(\widehat{\mathcal{L}}(\mathsf{U}_{i,n}))}(x).$$

The identity would give a fast computation of the Hilbert–Poincaré series of uniform matroid Chow rings, but the paper gives no resolution.