Non-removability conjecture for Sierpiński carpets

A Sierpiński carpet is a planar set homeomorphic to the standard Sierpiński carpet $S_3$, equivalently a compact set obtained from a Jordan region by removing countably many Jordan regions with pairwise disjoint closures, diameters tending to zero, and empty interior remaining. A set is removable for quasiconformal maps, respectively for $W^{1,p}$ functions, if every map or function with the relevant regularity off the set extends with that regularity across the set.

Sierpiński carpet non-removability conjecture. Every Sierpiński carpet is non-removable for quasiconformal maps and for $W^{1,p}$ functions, for

$$1\leq p\leq\infty.$$

The standard Sierpiński carpet is known to be non-removable in these settings, but the corresponding statement for arbitrary generalized Sierpiński carpets is not known in general.