Cohomology rings determining bunches of grapes

Determine whether the cohomology ring of the total configuration space of a bunch of grapes determines the underlying graph up to homeomorphism.

Background

The paper proves that, for a bunch of grapes, the sequence of Hilbert polynomials determines the multiset of local data at essential vertices. However, distinct non-homeomorphic bunches of grapes can share the same local data and therefore the same Hilbert polynomials. The authors contrast this with Sabalka’s result for trees, where the cohomology ring of the total configuration space characterizes the graph up to homeomorphism, and ask whether an analogous statement holds for bunches of grapes.