Extension conjecture for nice Browkin continued fractions

Let $p$ be an odd prime, let $\mathcal{Y}$ be the set of allowed partial quotients, and let $a_0,\ldots,a_{t-2}\in\mathcal{Y}$ satisfy $a_i>0$ and $|a_0|_\infty<\frac p4$. Extension conjecture for nice BCFs. There exists $a_{t-1}\in\mathcal{Y}$ such that the Browkin continued fraction $[a_0,\ldots,a_{t-1}]$ is nice. This is presented as a stronger experimental assertion intended to support the construction of periodic BCF expansions; the supplied status evidence records the relevant case as resolved by Bedocchi (1989).