Strong Markoff uniqueness conjecture

A Markoff triple is a positive integer solution $(x,y,z)$ of

$$x^2+y^2+z^2=3xyz.$$

For each Markoff triple $(x,y,z)$, let $\xi$ be the corresponding badly approximable irrational, whose field of definition is the quadratic field generated by $\sqrt{9z^2-4}$.

Strong Markoff uniqueness conjecture. Each of the badly approximable irrationals $\xi$ corresponding to a Markoff triple $(x,y,z)$ has a different field of definition $\mathbb{Q}[\sqrt{9z^2-4}]$. Equivalently, if $(x,y,z)$ and $(x',y',z')$ are distinct Markoff triples, then

$$\frac{9z^2-4}{9z'^2-4}$$

is not a square in $\mathbb{Q}$.

This strengthens Markoff uniqueness by requiring the quadratic fields attached to distinct Markoff triples to be distinct. The supplied text presents it as a proposed strengthening and gives no resolution.