Determine whether maximal-shattering numbers are non-decreasing

Determine whether the sequence (s_k(X,S))_{k∈N} is non-decreasing for every set system (X,S) with finite VC-dimension.

Background

The maximal-shattering sequence s_k(X,S) counts the isomorphism types of subsets of maximum size that are shattered by the k-fold union of a set system S. Although the sequence of corresponding VC-dimensions is non-decreasing because S is contained in each k-fold union, the paper notes that monotonicity of the number of isomorphism types does not follow from this observation. The authors explicitly ask whether the maximal-shattering sequence must nevertheless be non-decreasing for every finite-VC set system.