Four-domain additive inequality for the hyperbolic metric

Let $D_1,D_2\subsetneq\mathbb{C}$ be simply connected domains, let $z\in D_1\cap D_2$, and assume that $D_1\cup D_2$ is hyperbolic. The hyperbolic densities satisfy

$$\lambda_{D_1\cap D_2}(z)+\lambda_{D_1\cup D_2}(z)\leq\lambda_{D_1}(z)+\lambda_{D_2}(z).$$

This conjecture strengthens the endpoint question for the simply connected upper estimate and is an additive inequality involving the intersection and union of two domains. Its status is open.