Vanishing global quasiconformal Assouad spectrum dimension below one

Let $0\le \theta<1$ and let $E\subset\mathbb{R}^n$. Write $\dim_A^\theta(E)$ for the Assouad spectrum with parameter $\theta$, and let the global quasiconformal Assouad spectrum dimension of $E$ with parameter $\theta$ be the infimum of the dimensions $\dim_A^\theta(f(E))$ over quasiconformal self-maps $f$ of $\mathbb{R}^n$. For $\theta=0$, interpret $\dim_A^0(E)$ as the upper box-counting dimension $\overline{\dim_B}E$. Vanishing-dimension conjecture. If

$$\dim_A^\theta(E)<1,$$

then the global quasiconformal Assouad spectrum dimension of $E$ with parameter $\theta$ is equal to zero. This would extend the known vanishing results for global quasiconformal Hausdorff and Assouad dimensions to the Assouad spectrum. The statement is posed as a conjecture, and no resolution is supplied in the source.