Continuity and monotonicity conjecture for local dimensions of CIFS spectra

Let $F\subset[0,\infty)$ be the dimension spectrum of a CIFS. For $x\in F$, define the local Hausdorff dimension of $F$ at $x$ by

$$\dim_x(F)=\inf\{\operatorname{HD}(F\cap B(x,\epsilon)): \epsilon>0\}.$$

Local-dimension regularity conjecture. The map

$$x\mapsto\dim_x(F)$$

restricted to $x\in F$ is continuous and weakly decreasing, meaning nonincreasing. This conjecture predicts strong regularity for the local geometry of every CIFS dimension spectrum; the source provides no resolution.