Hensley’s A = 2 version of Zaremba’s conjecture for large denominators

Determine whether Zaremba’s bound can be strengthened to A = 2 for all sufficiently large denominators u, i.e., whether every sufficiently large integer u admits a coprime t with 1 ≤ t < u such that the continued fraction of t/u has all partial quotients bounded by 2.

Background

The standard form of Zaremba’s conjecture asserts the existence of a universal bounded A for all denominators. Hensley proposed the stronger form that A = 2 suffices beyond some threshold, which, if true, would represent a remarkable sharpening of the bounded partial quotient phenomenon in continued fractions.