Conjecture on nice Browkin continued fractions of every length

Let $p$ be a prime. A nice Browkin continued fraction is the special class of Browkin continued fractions introduced by Capuano, Murru, and Terracini. The nice-continued-fraction conjecture. For every integer $t\geq 1$, there exists a nice Browkin continued fraction of length $t$, except when $t=1$ and $p=3$. The conjecture concerns the existence of the finite starting sequences used to construct infinite families of quadratic irrationals with periodic Browkin expansions; the source leaves it as an open problem.