Sundaram–Welker acyclicity conjecture for partition-poset complexes

Let $\lambda$ and $\mu$ be different set partitions with $\lambda\vdash\mu$, and let $X_{\lambda,\mu}$ be the associated complex introduced in the paper. Here $\lambda\vdash\mu$ denotes the stipulated refinement relation between the set partitions.

Sundaram–Welker's acyclicity conjecture. For $\lambda\vdash\mu$ with $\lambda\neq\mu$, the space $X_{\lambda,\mu}$ is ${\mathbb Q}$-acyclic.

This is presented as equivalent to the original formulation concerning the multiplicity of the trivial representation in the stabilizer module on the reduced homology of the corresponding partition-poset interval. The supplied text gives no resolution status.