Diophantine Conjecture on bounded partial quotients for alternating-ones continued fractions of t/u

Determine whether there exists a universal constant A > 0 such that, for every integer t ≥ 3, there is a coprime integer u > t with the rational t/u admitting a continued fraction expansion of the form [b1, 1, b2, 1, …, bm, 1] in which all partial quotients b1, …, bm are bounded by A.

Background

Via the Main Graph Theorem, a continued fraction of the specific alternating-ones form [b1,1,b2,1,…,bm,1] yields a connected simple planar graph G with τ(G) = t and |V| roughly proportional to b2+…+bm. If one could ensure uniformly bounded b_i independent of t, the construction would imply α(t) = O(log t). This conjecture is a tailored variant of bounded-digit phenomena in continued fractions, closely related to Zaremba’s conjecture.