Real convergence of Browkin-type continued fractions

Let $\alpha^{(p)}$ be a quadratic irrational in $\mathbb{Q}_p$ that has an image in $\mathbb{R}$. Consider its continued fraction obtained by Browkin I, Browkin II or Algorithm MR. Browkin real-convergence conjecture. These continued fractions always converge in $\mathbb{R}$. This is presented as the Browkin-type analogue of the known real-convergence theorem for Ruban continued fractions and is supported by computational observations; it remains open.