Conjecture on even-period Browkin II expansions of square roots

Let $h\in\mathbb{Z}$ be even. Consider the $p$-adic continued fraction expansion produced by the Browkin II algorithm. Even-period Browkin II conjecture. For every even $h\in\mathbb{Z}$, there exist infinitely many nonsquare integers $D$ such that the Browkin II continued fraction expansion of $\sqrt{D}$ is periodic with period length $h$. The paper proves the existence of infinitely many examples for period length $4$ and recalls Browkin's result for period length $2$; experimental evidence suggests that the result may extend to all even period lengths, but the general claim remains open.