The contact process is a classical model for epidemic spreading on graphs, where each node is either infected or healthy, and infection spreads across edges at some rate while recovery happens independently. Recent work introduces social cluster structure, meaning nodes are grouped into small communities through which disease can spread internally before jumping between clusters. The open problem is whether the critical infection rate for global epidemic survival depends analytically on the cluster size distribution, and specifically whether one can derive a closed-form or asymptotically sharp formula for this critical threshold as cluster sizes grow. This includes understanding whether the phase transition remains sharp in the statistical mechanics sense when heterogeneous cluster structures are introduced alongside asymptomatic carrier states.