  For any twisted ideal polygon in $\mathbb{H}^3$, we construct a harmonic map
from $\mathbb{C}$ to $\mathbb{H}^3$ with a polynomial Hopf differential, that
is asymptotic to the given polygon, and is a bounded distance from a pleated
plane. Our proof uses the harmonic map heat flow. We also show that such a
harmonic map is unique once we prescribe the principal part of its Hopf
differential.
