Transcendental harmonic extension of the real non-attractive fixed point conjecture

Determine whether the real non-attractive fixed point conjecture extends to transcendental harmonic mappings f = h + \overline{g} in the complex plane where at least one of h or g is transcendental; specifically, ascertain whether such a function admits a \mathfrak{h}-fixed point \zeta = \mu + \overline{\omega} whose multipliers satisfy Re(h'(\mu)) \ge 1 and Re(g'(\omega)) \ge 1.

Background

The paper proves the real non-attractive fixed point conjecture for complex harmonic functions in two settings: rational harmonic functions with a super-attracting fixed point, and polynomial harmonic functions (for which \infty is super-attracting). In these cases, the authors show the existence of a \mathfrak{h}-fixed point \zeta = \mu + \overline{\omega} whose multipliers have real parts at least 1.

For transcendental harmonic mappings, where at least one of h or g is transcendental, the situation is unresolved. The authors highlight an example f(z) = e^z + \overline{z^2} for which no finite \mathfrak{h}-fixed points exist, underscoring that techniques used for polynomial or rational cases may not directly apply and motivating the explicit question of whether the conjecture holds in the transcendental setting.