A stronger form of countable dense homogeneity of the plane

Let C={C1,C2,…}, K={K1,K2,…} be countable families of subsets of the Euclidean plane R2 whose diameters tend to zero and whose closures are continua such that cl(Ci)∩cl(Cj)=∅ and cl(Ki)∩cl(Kj)=∅, for i≠j, i,j∈N. If all sets from both families C and K are ambiently homeomorphic to each other via orientation preserving automorphisms of R2, and the sets ⋃C, ⋃K are dense in R2, then there exists an automorphism f:R2→R2 of the plane such that {f[C]:C∈C}=K.