The distance-set dimension improvement conjecture

For a compact set $K\subset\mathbb{R}^2$, let $\operatorname{dist}(K)=\{|x-y|:x,y\in K\}$ be its distance set, and let ${\text{\roman dim}}$ denote Hausdorff dimension. Distance-set improvement conjecture. There exists an absolute constant $c_0>0$ such that

$${\text{\roman dim}}(\operatorname{dist}(K))\geq\frac12+c_0$$

whenever ${\text{\roman dim}}(K)\geq1$. This is weaker than Falconer's distance set conjecture and was stated as remaining open.