Spherical wedge form of the minimal spanning poset nerve

Determine whether the nerve N P^∘ of the poset of nontrivial minimal spanning posets in Sub_{R^k} has the homotopy type of a wedge of (2k−3)-spheres for a commutative ring R and free module R^k.

Background

The authors define P^∘ as the full subposet of Sub_{R^k} consisting of nontrivial minimal spanning posets (closed under intersections, with colimit R^k, and minimal under these conditions). They establish that D(R^k) ≃ Σ^∞ Σ N P^∘ and note that Σ N P^∘ is (2k−2)-dimensional.

Motivated by analogous decompositions for finite sets, inner product spaces, and free modules over fields/Dedekind domains where related posets have wedge-of-spheres homotopy types, they conjecture that N P^∘ itself is a wedge of spheres of dimension 2k−3.