A bounded globally univalent quasiconformal harmonic map whose analytic part is unbounded

We construct, for every $0<k<1$, a bounded globally univalent harmonic mapping $$f=h+\overline g \colon \D\to\C$$ such that $$|g'(z)|\le k|h'(z)|,\qquad z\in\D,$$ while the analytic part $h$ is unbounded. The construction is based on a bounded logarithmic spiral map on a far right horizontal strip, together with a smaller logarithmic perturbation.