The asymptotic bound conjecture for monochromatic progressions in the Fibonacci word

Let $A(d)$ denote the length of the longest monochromatic arithmetic progression of difference $d$ in the Fibonacci word, and let $\tau$ denote the golden ratio. The authors conjecture that

The asymptotic bound conjecture. For all $d\geq 1$,

$$\frac{A(d)-1}{d}<\frac{\sqrt{5}}{\tau}.$$

The conjecture is motivated by the hope that sufficiently strong bounds on families of $A(d)$ could improve the asymptotic estimates for these progression lengths; the paper states that this goal was not achieved.