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Descriptive complexity of embeddability on closed subsets of certain Polish spaces

Determine the descriptive set-theoretic complexity (e.g., Borel, analytic-complete) of the embeddability quasi-order on closed subsets of a Polish space X that contains the Cantor space but does not contain [0,1]^2.

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Background

Embeddability between closed subsets of [0,1]2 is known to be analytic complete, implying maximal complexity. For Polish spaces that include Cantor space but exclude [0,1]2, the complexity of embeddability is unclear. Clarifying this would impact understanding of reducibility and embeddability phenomena across different Polish spaces.

References

The same question is open for continuous functions on any uncountable Polish space that does not contain [0,1]2. In fact, the following question is already open:

Question Let X be a Polish space containing \cantor but not [0,1]2. What is the complexity of embeddability on closed subsets of X?

— A well-quasi-order for continuous functions (2410.13150 - Carroy et al., 17 Oct 2024) in Section 7.3 (DirectionPolish)