The maximal monotone mapping relative-boundary image conjecture

Let $F\colon\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ be a maximal monotone mapping. Then the maximal monotone mapping relative-boundary image conjecture. The image of the map $x\mapsto \operatorname{rb} F(x)$ has Lebesgue measure zero, that is, the set

$$\bigcup_{x\in\mathbb{R}^n} \operatorname{rb} F(x)$$

is Lebesgue null.

For maximal cyclically monotone mappings, the analogous statement follows from the corresponding result for closed proper convex functions, while the conjecture concerns arbitrary maximal monotone mappings.