The conjectured first-difference sequence for the Fibonacci word

Let $f$ be the infinite Fibonacci word, let $\Sigma$ be its alphabet, and define

$$M_n=\min\{\operatorname{Cl}(w):w\in\operatorname{Fac}(f)\cap\Sigma^n\},\qquad R_n=M_n-M_{n-1}.$$

Here $\operatorname{Cl}(w)$ is the number of distinct closed factors of $w$, and $F_n$ denotes the Fibonacci sequence used in the source. Fibonacci first-difference conjecture. The sequence $(R_n)_{n\geq1}$ is

$$1,1,1,1,F_0,F_2,F_0,F_2,F_1,F_1,F_3,F_1,F_1,F_3,F_2,F_2,F_2,F_4,F_4,F_2,F_2,F_2,F_4,F_4,\ldots,$$

with the displayed infinite product pattern specified in the source.

The conjecture is based on numerical experiments and would yield an explicit formula for $M_n$. The paper does not report a proof or a resolution.