Nonvanishing imaginary-part conjecture for reduced Beltrami solutions

Let $\Omega$ be a domain in $\mathbb C$, and let $f:\Omega\to\mathbb C$ be a quasiregular solution of the reduced Beltrami equation

$$\frac{\partial f}{\partial \bar z}=\lambda(z)\\,\operatorname{Im}\left(\frac{\partial f}{\partial z}\right), \qquad |\lambda(z)|\leq k<1.$$

Nonvanishing imaginary-part conjecture. Either $\operatorname{Im}(f_z)$ is constant, or

$$\operatorname{Im}\left(\frac{\partial f}{\partial z}\right)\ne 0$$

almost everywhere in $\Omega$. This conjecture concerns a proposed analogue of the familiar nonvanishing property of the Jacobian determinant for solutions of the reduced Beltrami equation; its status is not established in the supplied source.