The finite-or-periodic conjecture for the p-adic Jacobi–Perron algorithm

Let $m\geq 1$, and let $1,\alpha_0^{(1)},\ldots,\alpha_0^{(m)}$ be $p$-adic numbers that are linearly dependent over $\mathbb{Q}$. The finite-or-periodic conjecture. The $p$-adic Jacobi–Perron algorithm on input $(\alpha_0^{(1)},\ldots,\alpha_0^{(m)})$ is either finite or periodic. This conjecture concerns the unresolved behavior of the algorithm on $\mathbb{Q}$-linearly dependent inputs: the source reports no observed infinite, nonperiodic example and explicitly describes the conjecture as still open.