A series of operators in $L^2(\mathbb C)$ proportional to unitary ones

We prove that singular integral operators in $L^2(\mathbb C)$ defined by the formula
$$ Tf(z)=\int_\mathbb C\frac{(w(z)-w(\xi))^n}{(z-\xi)^{n+2}}f(\xi)\,dm_2(\xi), $$
where $|w(z)-w(\xi)|\leq c|z-\xi|$, $z,\xi\in\mathbb C,$ are proportional to unitary ones if and only if $w(z)=az$ or $w(z)= b\overline z$.