Hilbert embeddability of sub-half snowflakes

Determine whether there exists a parameter \(0<\theta<\tfrac12\) such that the snowflaked zero section \(((_2\wr^2)_0,d_{_2\wr^2}^{\theta})\) admits a bi-Lipschitz embedding into \(\ell_2\).

Background

The paper rules out the analogous bi-Lipschitz embedding for snowflake exponents between $1/2$ and $1$, using the established Lq distortion estimates and the isometric embedding of $\ell_2$ into $L_q$. It leaves open the range of exponents strictly below $1/2$, formulating the question for the zero section of the planar lamplighter group.