Baril–Kirgizov–Vajnovszki conjecture on 1-Gray codes for restricted binary words

Let $\mathcal{W}_{q,n}$ denote the set of restricted binary words of length $n$ associated with the positive rational parameter $q$. A 1-Gray code is an arrangement of all words in which consecutive words differ in at most one position. Baril–Kirgizov–Vajnovszki conjecture. For given $q \in \mathbb{N}^+$ and $n \ge 0$, a 1-Gray code exists for $\mathcal{W}_{q,n}$. This extends the known existence of a 3-Gray code and is motivated by the Eğecioğlu–Iršič conjecture, proved for $q=1$; the assertion is based on experimental evidence for small values of $n$ and $q$ and remains open in the supplied source.