The conjectural formulas for earliest Thue–Morse progressions in particular families

Let ${\mathbf t}$ be the Thue–Morse word. For a positive integer difference $d$, let $A(d)$ be the length of the longest monochromatic arithmetic progression of difference $d$, and let $i(d)$ be the starting position of the first such progression of length $A(d)$. For the infinite families of differences considered below, the authors conjecture that

The conjectural formulas for earliest Thue–Morse progressions.

$$i(2^n+1)=3\cdot 2^{2n}-2^n-1,$$ $$i(2^{2n}-1)=3\cdot 2^{4n}-2^{2n}+1,$$ $$i(2^{2n+1}-1)=2^{2n+1}-1.$$

These identities are supported by empirical observations and are stated after known formulas for the corresponding values of $A(d)$, but a proof remains elusive.