Assouad-dimension analogue of the inf-harmonicity theorem

Establish whether, for every holomorphic motion f of the Riemann sphere and every bounded subset E of the complex plane, either the Assouad dimension dim_A(f_λ(E)) is zero for every parameter λ in the unit disk or the function λ ↦ 1/dim_A(f_λ(E)) is inf-harmonic on the unit disk.

Background

The paper proves that the reciprocal of the quasi-Assouad dimension of the image of a bounded set under a holomorphic motion is inf-harmonic, yielding quasiconformal distortion bounds for quasi-Assouad dimension. Since quasi-Assouad dimension is bounded above by Assouad dimension, the authors ask whether the same parameter-variation result holds for the stronger Assouad dimension.

The authors explicitly state that they were unable to prove this analogue. They note separately that the corresponding quasiconformal distortion inequality for Assouad dimension is already known, so the unresolved issue is specifically the inf-harmonicity characterization under arbitrary holomorphic motions.