The roots-of-unity and irrational-angle conjecture for complex Markov triples

Let $(z_1,z_2,z_3)$ be a triple of complex roots of unity, and let $n$ be the minimal positive integer such that $z_i^n=1$ for each $i$. Write the arguments as fractions of a full turn. Complex Markov argument conjecture. Every triple of fractions $\frac{a}{n},\frac{b}{n},\frac{c}{n}$ whose sum is $1$ occurs as the arguments of a complex Markov triple generated from $(z_1,z_2,z_3)$. Furthermore, if the initial triple has irrational angles, then the image of the arguments of Markov triples is dense in the circle. This conjecture describes the possible argument dynamics separately from the classical norm dynamics; the source gives examples and poses the assertion without a resolution.