The generator-number conjecture for orbit levels

Let $G$ be a group generated by $n$ homeomorphisms, and let the level of an orbit mean the level classification used for the orbit structure in the source.

Generator-number conjecture. The highest possible level of an orbit is $n+1$.

This conjecture predicts a direct bound on orbit complexity in terms of the number of homeomorphisms generating the group. The source presents it as a further expected statement and gives no resolution.