Conjecture: infinitude of the unit-circle intersection with K_λ × K_λ for 2 − √3 < λ < 1/2

Prove that for every parameter λ with 2 − √3 < λ < 1/2, the intersection of the unit circle S = {(x, y) ∈ R^2 : x^2 + y^2 = 1} with the Cartesian product K_λ × K_λ is infinite, where K_λ ⊂ [0,1] is the self-similar set generated by the iterated function system {f_0(x) = λx, f_1(x) = λx + 1 − λ}.

Background

The paper proves a trichotomy for S ∩ (K_λ × K_λ): it is trivial for 0 < λ ≤ 2 − √3, non-trivial for λ ≥ 0.330384, and of continuum cardinality for λ ≥ 0.407493. It is also shown that the threshold 2 − √3 is sharp for triviality.

Despite these bounds, the exact behavior for all λ in the interval (2 − √3, 1/2) remains unsettled. The authors explicitly conjecture that the intersection is infinite throughout this entire range, which would close the gap between the sharp triviality threshold and the current non-triviality/continuum guarantees.