Semi-proximality of ladder systems under R-embeddability with no Cantor copy in the image of ladder points

Prove that a ladder system L on ω1 is semi-proximal whenever there exists a continuous embedding H: Y(L) -> R (equivalently, 2^ω) witnessing R-embeddability such that H[Lim × {1}] contains no copy of the Cantor set.

Background

In Section 4 the authors show two contrasting results: if L is a **-sequence then Y(L) is not semi-proximal (Theorem 1), while every uniformizable ladder system is semi-proximal (Proposition 5). They also note that uniformizable ladder systems are R-embeddable (Proposition 6).

Motivated by these results, they conjecture a sufficient condition for semi-proximality of ladder systems based on R-embeddability with the additional requirement that the image of the ladder-point set Lim × {1} contains no copy of the Cantor set. This would extend Proposition 5 to a broader class beyond uniformizable ladder systems.