Three-type classification conjecture for CIFS dimension spectra

Let $F\subset[0,\infty)$ be the dimension spectrum of a CIFS, and let $\dim_x(F)$ denote its local Hausdorff dimension at $x\in F$. Three-type classification conjecture. Exactly one of the following mutually exclusive alternatives holds:

1. Type I: $F$ is a finite union of intervals.
2. Type II: $F$ has Hausdorff dimension zero.
3. Type III: There is a constant $c$ with $0<c<\sup(F)$ such that

$$\dim_x(F)=\min(1,c/x)\qquad\text{for all }x\in F.$$

In Type III, the graph of $x\mapsto\dim_x(F)$ is a horizontal line followed by a hyperbola. This conjecture aims to classify all possible topological and metric forms of CIFS dimension spectra; the source provides no resolution.