Partition and almost disjoint properties of PP-rich sets in uncountable semigroups

Determine whether PP-rich sets retain partition and almost disjoint properties when the underlying semigroup S is uncountable.

Background

For PP-rich subsets of (N, +), the paper proves that every such set contains 2^ω almost disjoint PP-rich subsets and can be split into countably many PP-rich subsets. The authors do not resolve whether analogous results hold for PP-rich sets in uncountable semigroups.

This question is distinct from partition regularity: it asks about the stronger infinite partition and almost disjoint conclusions in the uncountable setting.