Existence conjecture for nice Browkin continued fractions

Let $p$ be an odd prime, let $\mathcal{Y}$ denote the set used in the paper for the allowed partial quotients, and let a nice BCF be a finite Browkin continued fraction satisfying the niceness conditions defined earlier in the paper. Existence conjecture for nice BCFs. For every integer $t\geq 1$, there exists a nice BCF of length $t$, except when $t=1$ and $p=3$. Nice sequences are the input to the paper's construction of infinitely many quadratic irrationals with periodic BCF expansions of period $2t$. The statement is marked resolved in the supplied status evidence: the exceptional $t=1$ case was proved in Bedocchi (1989).