Quasiconvexity equals rank-one convexity for isotropic sets of 2x2 matrices

Let K be a given compact set of real 2x2 matrices that is isotropic, meaning invariant under the left and right action of the special orthogonal group. Then we show that the quasiconvex hull of K coincides with the rank-one convex hull (and even with the lamination convex hull of order 2). In particular, there is no difference between quasiconvexity and rank-one convexity for K. This is a generalization of a known result for connected sets