Commutativity preserving transformations on conjugacy classes of compact self-adjoint operators

Let $H$ be a complex Hilbert space of dimension not less than $3$ and let ${\mathcal C}$ be a conjugacy class of compact self-adjoint operators on $H$. Suppose that the dimension of the kernels of operators from ${\mathcal C}$ not less than the dimension of their ranges. In the case when ${\mathcal C}$ is formed by operators of finite rank $k$ and $\dim H=2k$, we assume that $k\ge 4$. We show that every bijective transformation of C preserving the commutativity in both directions is induced by a unitary or anti-unitary operator up to a permutation of eigenspaces of the same dimension.