Existence of 1-Gray codes for Fibonacci q-decreasing words

Let $n\geq 0$ and $q\geq 1$, and let $\mathcal{W}^q_n$ denote the set of Fibonacci $q$-decreasing words of length $n$. A 1-Gray code for $\mathcal{W}^q_n$ is an ordering of its words in which consecutive words differ in exactly one position. The 1-Gray code conjecture. For any $n\geq 0$ and $q\geq 1$, there is a 1-Gray code for $\mathcal{W}^q_n$. The conjecture extends the established construction for the case $q=1$; the paper notes that the parity condition and experiments for $0\leq n\leq 5$ and $2\leq q\leq 5$ support it, but no general construction is given.