Extension of the word-function observations for Hecke groups

Let $G_k$ be the Hecke group with parameter $k$, let $W$ denote the words under consideration, and let $\nu_1,\ldots,\nu_8$ be the functions defined in Observation 1, using the Fibonacci numbers $F_n$ and the relations $\ll_1,\ldots,\ll_4$ specified there. Observation 1. Observation 1 holds for words of arbitrary length and all $k\geq 5$. The claim extends experimentally observed formulas for the functions represented by words in $G_k$; the supplied text gives no proof or resolution.