The Fibonacci-index formula conjecture for earliest progressions

Let $F_n$ denote the Fibonacci numbers. For a positive integer difference $d$, let $i(d)$ be the starting position of the first monochromatic arithmetic progression of difference $d$ having maximum length. Empirical evidence suggests that

The Fibonacci-index formula conjecture.

$$i(F_{2n+1})=F_{2n+3}-2,$$

and

$$i(F_{2n})=F_{4n}-1.$$

The formulas concern the case $d=F_n$, where the corresponding longest-progression lengths $A(F_n)$ are not eventually constant. The paper explains that this prevents the authors from checking the identities automatically with their stated method, so the conjecture remains open.