Conjecture that the dilatation condition can be removed from the spherical-area theorem

Let $f=h+\overline g\in\mathcal S_H^0$ be a normalized univalent, sense-preserving harmonic mapping, and let $\Omega=f(\mathbb D)$. Theorem 2 in the paper proves the stated spherical-area comparison under a bound on the dilatation. Dilatation-removal conjecture. The condition on the dilatation in that theorem can be removed. This would extend the spherical-area comparison from mappings satisfying the theorem's dilatation hypothesis to all mappings covered by its other assumptions; the source states the claim but gives no resolution.