Conjecture on successive-step ratios for Thue–Morse and Zeckendorf complexity

For a sequence $\mathcal{S}$, let a step be an integer $N$ at which its maximum order complexity changes from $M(\mathcal{S},N-1)$ to $M(\mathcal{S},N)$, and define the ratio of successive steps using consecutive steps with no intermediate complexity value. Let $\varphi=(1+\sqrt{5})/2$. Successive-step ratio conjecture. The ratio of successive steps tends to $2$ for a sequence related to the Thue–Morse sequence and tends to $\varphi$ for the sequence related to the Zeckendorf sum of digits.

This conjecture concerns the observed asymptotic spacing of complexity jumps and is supported by numerical experimentation; no proof or resolution is supplied in the paper.