Non-removability conjecture for planar compact sets containing a Cantor strip

Let $C$ be the middle-thirds Cantor set. A planar compact set $K$ is said to contain a homeomorphic copy of $C\times[0,1]$ if there is a subset of $K$ homeomorphic to $C\times[0,1]$. A compact set is non-removable for conformal maps if there exists a homeomorphism of the Riemann sphere that is conformal off the set but is not a Möbius transformation. Non-removability conjecture. Every planar compact set containing a homeomorphic copy of $C\times[0,1]$ is non-removable for conformal maps. This conjecture generalizes the preceding result that all Sierpiński carpets are non-removable for conformal maps, and remains open in the source.