Simultaneous real and p-adic convergence implies periodicity

Let $[a_0,a_1,\ldots]$ be a continued fraction with $a_n\in\mathbb{Q}$ for all $n$. Suppose it converges simultaneously in $\mathbb{R}$ and $\mathbb{Q}_p$ to a real quadratic irrational $\alpha^{(r)}$ and its image $\alpha^{(p)}$ in $\mathbb{Q}_p$, respectively. Simultaneous-convergence conjecture. The continued fraction $[a_0,a_1,\ldots]$ is periodic. This broad formulation is proposed alongside a Browkin-specific version and is supported by the paper's experimental evidence; no proof or resolution is given.