Falconer's conjecture on stable intersections of regular Cantor sets

Let $K\subset\mathbb{R}^d$ be a regular Cantor set, and write $\operatorname{\dim_{H}}(K)$ for its Hausdorff dimension. A pair of sets has a $C^r$-stable intersection if this intersection persists under sufficiently small $C^r$ perturbations, for some $r>1$. Falconer's conjecture. Every regular Cantor set $K\subset\mathbb{R}^d$ with

$$\operatorname{\dim_{H}}(K)>d/2$$

can be $C^\infty$-approximated by a regular Cantor set $\tilde K$ such that the pair $(\tilde K,\tilde K)$ has a $C^r$-stable intersection for some $r>1$. This is a critical-case formulation of the stable intersection problem. Falconer's conjecture remains open even for $d=2$, where the best-known dimension threshold is $5/4$.