Co-primeness conjecture for odd higher-dimensional q-discrete Painlevé II equations

Let $k=2m+1\geq 7$ be odd, and define rational functions $y_n\in\mathbb{Q}(y_0,\dots,y_{2m-3},\alpha,\beta,\gamma)$ by the difference equation specified in the source. Odd-dimensional co-primeness conjecture. If $|n-n'|>k-2$, then $y_n$ and $y_{n'}$ are co-prime. This conjecture extends the proved co-primeness result for $k=5$ to all odd $k\geq 7$; its status is not resolved in the supplied text.