Lq distortion growth beyond the Hilbert range

Determine the Lq distortion growth of the zero section \(((_2\wr_n^2)_0,d_{_2\wr_n^2}^{1/q})\) for \(2<q<\infty\), including whether its distortion can be bounded by an absolute constant for any such q.

Background

For $1\le q\le 2$, the paper establishes that the relevant distortion is of order $(\log n)^{1/q}$. The authors explicitly state that they do not know whether this formula extends to $q>2$, and specifically do not know whether constant distortion occurs for any exponent in that range.