Dekking's conjecture on the subword complexity of the Fibonacci-Thue-Morse sequence

Let $({\bf ftm}[n])_{n\geq 0}$ be the Fibonacci-Thue-Morse sequence, obtained by summing modulo $2$ the bits in the Fibonacci representation of $n$. Let $\rho_{\bf ftm}(n)$ denote its subword complexity, and define

$$d(n):=\rho_{\bf ftm}(n+1)-\rho_{\bf ftm}(n).$$

For each $i\geq 2$, let $F_i$ denote the $i$th Fibonacci number. Dekking's conjecture. The first-difference sequence of the subword complexity is

$$(d(n))_{n\geq 0}=(1,2,4,6,10)\prod_{i\geq 2}6^{F_i+(-1)^i}\,8^{F_i},$$

that is, it begins

$$1,2,4,6,10,6,6,8,6,8,8,6,6,6,6,8,8,8,6,6,6,\ldots.$$

The quantity $d(n)$ counts the right-special factors of length $n$ in the Fibonacci-Thue-Morse sequence. The paper proves this conjecture, so the asserted formula is no longer open.