The effective Lang-type approximation conjecture for cubic irrational numbers

Let $\xi\in\mathbb{R}$ be a real cubic irrational number, let $H(\xi)$ be the naive height of its minimal polynomial with coprime integer coefficients, and let $\varphi=(\sqrt{5}+1)/2$. Let $\tau$ and $C$ be the constants from the partial-quotient bound conjecture. Effective Lang-type approximation conjecture. There exists an absolute constant $c$ such that, for every real cubic irrational number $\xi$ and every $q\in\mathbb{N}$,

$$\lVert q\xi\rVert\geq\frac{c}{H(\xi)^\tau q(\log q)^2}.$$

Moreover, one can take $c=\frac{\ln^2\varphi}{C}$. This is presented as a reformulation of the preceding conjecture using standard continued-fraction estimates. The supplied text gives no independent resolution of the claim.