The conjectural classification of Markoff-Rosenberger triples in arithmetic progression over quadratic fields

Let $d$ range over ${\mathbf Z}/{\mathbf Z}^*$ and let $D$ range over ${\mathbf Z}/{\mathbf Z}^2$. For a quadratic field ${\mathbf Q}(\sqrt D)$, write ${\mathcal A}{\mathcal P}_{(1,1,1,d)}({\mathbf Q}(\sqrt D))$ for the set of Markoff triples in arithmetic progression with parameter $d$.

Classification conjecture.

$$\#\left[\bigcup_{(d,D)\in {\mathbf Z}/{\mathbf Z}^*\times {\mathbf Z}/{\mathbf Z}^2}{\mathcal A}{\mathcal P}_{(1,1,1,d)}\left({\mathbf Q}(\sqrt D)\right)\right]=178.$$

Moreover, the displayed tables give all triples in arithmetic progression over quadratic fields; if $(d,D)$ does not occur in the tables, then

$${\mathcal A}{\mathcal P}_{(1,1,1,d)}\left({\mathbf Q}(\sqrt D)\right)={\mathcal A}{\mathcal P}_{(1,1,1,d)}({\mathbf Q}).$$

The conjecture summarizes computations for real quadratic fields and proposes a complete explicit classification of the Markoff triples in arithmetic progression over all quadratic fields. The paper provides the listed examples and counts but does not establish the classification in general.