Mirror-spectrum interval conjecture for CIFSes

Let $K\subset[0,\infty)$, and write its mirror image as $\sup(K)-K=\{\sup(K)-x:x\in K\}$. A dimension spectrum is a set arising as the set of Hausdorff dimensions of the limit sets of all subsystems of a CIFS. Mirror-spectrum interval conjecture. If both $K$ and $\sup(K)-K$ are dimension spectra of CIFSes, then there is some $\lambda>0$ such that

$$K=[0,\lambda].$$

This proposes that simultaneous realization of a set and its reflection is possible only for intervals; the source gives no evidence of resolution.