Lifting-length conjecture for Markoff triples

Let $G_p$ be the graph of Markoff triples modulo a prime $p$, and let a triple in $G_p$ be lifted to a triple in the Markoff tree over $\mathbb Z$. The lifting-length conjecture states that the length of a path found by such a lift is at least

$$\operatorname{O}(\log p)$$

for most triples in $G_p$.

The conjecture concerns the potential complexity of lifting attacks on the Markoff-triple hash function; the source gives no resolution or further evidence for it.