Non-invertible case (δ_L<0, δ_R>0): 2^n-component structure in R^{(4)}_n

Prove that for every integer n ≥ 1 and every parameter ξ in R^{(4)}_n = { ξ ∈ Φ^{(4)} | φ_min(g^n(ξ)) > 0, φ_min(g^{(n+1)}(ξ)) ≤ 0, α(ξ) < 0, α(g(ξ)) < 0 }, the border-collision normal form f_ξ has a chaotic attractor with exactly 2^n connected components.

Background

To avoid regions with stable period-four solutions (LRRR cycles), the authors impose α(g(ξ)) < 0 in R^{(4)}_n. Using renormalisation and numerical evidence, they posit a 2^n-component structure analogous to orientation-preserving and other non-invertible cases.

A full proof across all n ≥ 1 is not provided; the following sentence explicitly frames this as a conjecture backed by numerics.