A boundary growth conjecture for harmonic mappings into linearly connected domains

Let $f=h+\bar{g}\in\mathcal{S}_{H}^{0}$ be a normalized locally univalent harmonic mapping, and let $\Omega=f(\mathbb{D})$ be an $M$-linearly connected domain. Write $\Lambda_f$ for the quantity used in the stated boundary estimate. Boundary growth conjecture. There is a positive constant $c_4<2$ such that, for $\xi\in\partial\mathbb{D}$ and $0\leq\rho\leq r<1$,

$$\Lambda_f(r\xi)\geq\frac{1}{8}\Lambda_f(\rho\xi)\left(\frac{1-r}{1-\rho}\right)^{c_4-1}.$$

This proposed estimate further refines the theorem's boundary growth inequality for mappings whose image is linearly connected; the supplied text gives no resolution.