Frobenius's unicity conjecture for Markoff numbers

Let $G_M$ be the graph whose vertices are the positive Markoff numbers, with edges arising from the Markoff transformations between coordinates of solutions to $x^2+y^2+z^2=3xyz$. Frobenius's unicity conjecture. The graph $G_M$ is a tree. The graph is known to be connected because every positive solution reduces to $(1,1,1)$, but the absence of cycles remains intractable; partial results are known.