Conjecture on occurrence sequences in base-phi expansions

Let $\beta(N)=\beta^+(N)\cdot\beta^-(N)$ be the base-phi expansion of the number $N$. Let $w$ be a word of length $m$, and let $R_{\cdot w}$ be the sequence of occurrences of numbers $N$ whose first $m$ digits of $\beta^-(N)$ are $w$, namely

$$d_{-1}\dots d_{-m}(N)=w.$$

Let $V_{\rm F}$, $V_{\rm G}$, and $V_{\rm H}$ denote the families of sequences whose first differences are respectively the fixed points of the morphisms $f$, $g$, and $h$, with arbitrary integer first element. Occurrence-sequence conjecture. There exist two Lucas numbers $a$ and $b$ such that $R_{\cdot w}$ is one of $V_{\rm F}$, $V_{\rm G}$, or $V_{\rm H}$. Alternatively, $R_{\cdot w}$ is a union of three sequences of these types. This conjecture proposes a uniform description of occurrence sequences arising from prefixes of the negative part of base-phi expansions; the paper's preceding examples motivate the three fixed-point families, but the supplied text gives no resolution.