Characterize semi-proximality for R-embeddable almost disjoint families via absence of perfect sets in the embedding image

Establish whether an R-embeddable almost disjoint family A on a countable set is semi-proximal if and only if the image of some (equivalently, every) continuous embedding f: Y(A) -> 2^ω (or R) restricted to A contains no uncountable perfect subset (equivalently, no copy of the Cantor set).

Background

Proposition 2 provides a sufficient condition: if there exists a continuous map h: Y(A) -> 2^ω with h|A injective and h[A] contains no perfect subset, then Y(A) is totally semi-proximal. This raises the question of whether the converse also holds.

The authors explicitly state that they do not know the converse and formulate a conjectural if-and-only-if characterization in terms of the absence of an uncountable perfect subset (a Cantor set) in the image of an embedding of Y(A). This would yield a precise topological-combinatorial criterion for semi-proximality of R-embeddable almost disjoint families.