Thue–Morse first-occurrence formulas for specified differences

Prove the conjectured formulas for the first starting positions i(d) of maximum-length monochromatic arithmetic progressions in the Thue–Morse sequence for d=2^n+1 and d=2^n−1, namely i(2^n+1)=3·2^{2n}−2^n−1, i(2^{2n}−1)=3·2^{4n}−2^{2n}+1, and i(2^{2n+1}−1)=2^{2n+1}−1.

Background

The paper studies not only the maximum length A(d) of monochromatic arithmetic progressions but also i(d), the earliest starting position of a progression attaining that maximum length. Earlier work had established formulas for A(2^n+1) and A(2^n−1), but the corresponding first-occurrence positions were supported only by empirical observations.

The authors state that the identities in the conjecture appear to hold, but that a proof remains elusive. Thus the problem is to establish these three exact formulas uniformly over the indicated integer parameters.