Equivalence conjecture for triples of complex numbers

Let $\sim$ be the equivalence relation on $m$-tuples of distinct complex numbers defined by $\Lambda\sim\Lambda'$ if $\mathcal{D}_n(\Lambda)=\mathcal{D}_n(\Lambda')$ for every natural number $n$. For triples, every tuple is equivalent to $(0,1,\lambda)$ with $\lambda\neq 0,1$; two such parameters are already known to give equivalent triples when they are both transcendental or when they are conjugate algebraic numbers.

Equivalence conjecture. If $(0,1,\lambda)\sim(0,1,\lambda')$, then either $\lambda$ and $\lambda'$ are both transcendental, or $\lambda$ and $\lambda'$ are conjugate algebraic numbers.

This conjecture seeks to characterize the equivalence classes of triples and remains open even for $m=3$. The stated equivalence relation is invariant under affine transformations and automorphisms of $\mathbb{C}$ over $\mathbb{Q}$, which explains the known cases.