On Meeks' conjecture on disjoint minimal graphs

We study disjoint solution pairs $(u,D)$ to the minimal surface equation in $\mathbb{R}^n$, with $u=0$ on $\partial D.$ A conjecture of W. Meeks states that at most two such pairs can exist in $\mathbb{R}^n.$ We confirm the conjecture in two complementary cases. For $n\in\{2,3,4,5,6\},$ we prove it