The even-period conjecture for Browkin II continued fractions

Let $p$ be a prime, and let $D\in\mathbb{Z}$ be not a perfect square. A Browkin II continued fraction is a continued fraction generated by Browkin's second $p$-adic continued-fraction algorithm. The Browkin II even-period conjecture. For every even integer $h$, there exist infinitely many $D\in\mathbb{Z}$ that are not perfect squares such that the Browkin II continued fraction of $\sqrt{D}$ is periodic with period length $h$. This is the analogue for Browkin II of Bedocchi's conjecture for Browkin I, extending the known existence of infinitely many examples with period length $4$; the source records it as a conjecture supported by experimental computations.