Bostan–Weil–Yurkevich conjecture on integral Pochhammer transforms

Let $(c_n)_{n\geq 0}$ be the sequence discussed in the paper, and for $(u,v)\in\mathbb{Q}^2$ define $\widetilde c_n=w^n(u)_n(v)_n c_n$, where $(u)_n$ and $(v)_n$ are Pochhammer symbols and $w\in\mathbb{Z}$. Bostan–Weil–Yurkevich conjecture. The only pairs $(u,v)\in\mathbb{Q}^2\cap(0,1)^2$ for which some such $w$ makes $\widetilde c_n\in\mathbb{Z}$ for every $n\geq0$ are those listed in the paper's figure; moreover, for every listed pair, $\sum_{n\geq0}\widetilde c_nx^n$ is algebraic with the algebraicity degree given in that figure. This conjecture concerns which rational Pochhammer factors turn the coefficients into integers and predicts algebraicity of all corresponding generating functions. The supplied status evidence explicitly says that it is an open problem.