Iversen-type conjecture for planar harmonic functions

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be harmonic. A cluster point of $f$ is a point in the cluster set of $f$ at infinity, and an asymptotic value is a value approached by $f(z)$ along a curve tending to infinity. The critical set of $f$ is the set where the differential of $f$ is singular.

Iversen-type conjecture. If the critical set of $f$ is nowhere dense and the cluster set of $f$ has empty interior, then every cluster point of $f$ is an asymptotic value. In particular, if every value in $f(\mathbb{C})$ is taken at most finitely many times, then every cluster point of $f$ is an asymptotic value.

This proposes an analogue of Iversen's theorem for planar harmonic functions, intermediate between the corresponding results for meromorphic or quasimeromorphic functions and the weaker alternatives available for continuous functions. The supplied text presents it as an expectation based on examples, and gives no resolution.