Topology of parameter regions

Determine whether, for every downward closed graph G in DC_N, the parameter region P_G is connected and whether it is simply connected.

Background

The parameter space is partitioned into regions P_G indexed by downward closed graphs, equivalently by Dyck paths. The paper proves that every such region has non-empty interior and describes the adjacency structure of their boundaries, but it does not settle the global topology of an individual region. The authors explicitly ask whether each region is connected or even simply connected.