Partition and almost disjoint properties of combinatorially rich sets in uncountable semigroups

Determine whether, for an infinite semigroup S of cardinality κ and a combinatorially rich subset A of S, the set A contains κ pairwise disjoint combinatorially rich subsets and, whenever κ contains λ almost disjoint subsets, contains λ almost disjoint combinatorially rich subsets.

Background

The paper proves that every combinatorially rich subset of (N, +) contains 2^ω almost disjoint combinatorially rich subsets and can be partitioned into countably many pairwise disjoint combinatorially rich subsets. The authors explicitly leave open whether the corresponding cardinal-generalized conclusions hold in arbitrary infinite semigroups, particularly uncountable ones.

The unresolved issue has two parts: obtaining κ pairwise disjoint combinatorially rich subsets from a combinatorially rich set of a semigroup of size κ, and obtaining λ almost disjoint such subsets whenever the cardinal κ itself supports a family of λ almost disjoint subsets.