Effective Lang conjecture for partial quotients of cubic irrationals

Let $x$ be a real cubic irrational number, let $H(x)$ denote the naive height of its minimal polynomial with integer coprime coefficients, and write its continued fraction as $[a_0;a_1,a_2,ots]$. Set

$$\tau=3+\frac{2\ln 2}{2.88}\approx 3.4814.$$

Effective Lang conjecture. There exists an absolute constant $C$ such that, for every real cubic irrational number $x$, all partial quotients satisfy

$$|a_n|\leq Cn^2H(x)^\tau.$$

This is presented as heuristic evidence toward an effective version of Lang's conjecture for cubic irrationals, supported partly by numerical computations and partly by the assumption that all unusually good rational approximations arise from convergents of the continued fractions considered in the paper. Its status is not resolved in the supplied text.