Conjugate functions of several variables in the class $\operatorname{Lip}_\alpha$

It is known that if a function $f$ of a single variable belongs to the class $\operatorname{Lip}(\alpha,C(\mathbf T))$ $(0<\alpha<1)$, then its conjugate function also belongs to the same class; in other words, the class $\operatorname{Lip}(\alpha,C(\mathbf T))$ $(0<\alpha<1)$ is invariant with respect to the operator of conjugation acting in it. In the two-dimensional case the class $\operatorname{Lip}(\alpha,C(\mathbf T^2))$ $(0<\alpha<1)$ is no longer invariant with respect to conjugate functions of two variables. Here a final result elucidating the full character of violation of invariance of the class $\operatorname{Lip}(\alpha,C(\mathbf T^N))$ $(0<\alpha<1)$ with respect to the multidimensional conjugation operator acting in it is established.