Infinitely many quadratic irrationals with every admissible even BCF period

Let $p$ be a prime and let $t\geq1$, excluding the case $p=3$, $t=1$. A Browkin continued fraction (BCF) is a $p$-adic continued fraction expansion, and $\mathcal{Y}$ is the set of allowed partial quotients. Even-period existence conjecture. There are infinitely many integers $p^{2k}m$, with $k\geq1$, $m\in\mathbb{Z}\setminus p\mathbb{Z}$ non-square, such that $\alpha=\frac1{p^k\sqrt m}$ has a BCF of the form

$$[a_0,\overline{a_1,\ldots,a_{t-1},a_t,a_{t-1},\ldots,a_1,2a_0}],$$

and $a_t=2c$ with $c\in\mathcal{Y}$. In particular, there are infinitely many $b\in\mathbb{Z}$ such that the BCF expansion of $\sqrt b$ has period $2t$. The claim is stated as a consequence that would follow from any of the preceding nice-sequence assertions; its supplied status is resolved, with the source attributing the relevant case to Bedocchi (1989).