Existence in ZFC of o-homogeneous spaces without pairwise disjoint witnesses

Establish, in ZFC, the existence of a zero-dimensional o-homogeneous separable metrizable space that is not o-homogeneous with pairwise disjoint witnesses, or determine whether such examples require additional set-theoretic assumptions.

Background

Section 4 shows that zero-dimensional analytic spaces admit o-homogeneity with pairwise disjoint witnesses (of optimal higher complexity). This invites the sharper question of whether there exist o-homogeneous spaces that nevertheless fail to admit pairwise disjoint homogeneous decompositions.

The authors point to this question as still open, asking for existence in ZFC and, failing that, under additional set-theoretic assumptions.