Maximal antichains below purity for abelian groups

Establish whether \(\mathscr{L}^{1}_{\mathbb{Z}}\), the sublattice of strong submodel relations on abelian groups below purity, contains an antichain of cardinality \(2^{2^{\aleph_0}}\).

Background

The paper constructs an uncountable antichain in $\mathscr{L}^{1}_{\mathbb{Z}}$ using an almost disjoint family of infinite sets of primes. It also proves that $$|\mathscr{L}^{1}_{\mathbb{Z}}|\leq 2^{2^{\aleph_0}}$$.

The open problem asks whether this upper bound is attained by the size of an antichain, making it the largest possible antichain cardinality in the lattice.