Classify zero-dimensional compact metrizable spaces homeomorphic to their squares when only isolated points occur outside the perfect kernel

Determine the number of homeomorphism types of compact metrizable zero-dimensional spaces X such that X is homeomorphic to X × X and every point of X is either isolated or belongs to the perfect kernel PK(X).

Background

The paper constructs, for each infinite M ⊆ ω \ {0}, a compact metrizable zero-dimensional space X(M) homeomorphic to its square, and shows that these spaces are pairwise non-homeomorphic, yielding continuum many examples.

Motivated by this construction, the authors consider refining the classification by bounding the Cantor–Bendixson ranks of points outside the perfect kernel. They pose a general question for each n ∈ ω about how many such spaces exist, noting that even the base case n = 0—where all non-kernel points are isolated—remains unsettled.

References

The situation is unclear even for $n=0$, i.e. we have the following question. How many compact metrizable zero-dimensional spaces $X$ are there (up to homeomorphism) such that $X$ is homeomorphic to $X \times X$ and each point in $X$ is either isolated or belongs to the perfect kernel of $X$?

— Compact spaces homeomorphic to their respective squares  (2401.07633 - Dudák et al., 2024) in Remark preceding the final Question, Section 4 (Zero-dimensional case)