The maximum-average-degree partition conjecture

Let $G$ be a graph, let $c_1,c_2$ be positive real numbers, and let $\mathrm{mad}(G)$ denote the maximum average degree of $G$. A partition $V(G)=A\uplus B$ divides the vertex set into disjoint parts, with $G[A]$ and $G[B]$ the induced subgraphs on those parts.

Maximum-average-degree partition conjecture. If

$$\mathrm{mad}(G)<c_1+c_2,$$

then there exists a partition $V(G)=A\uplus B$ such that

$$\mathrm{mad}(G[A])<c_1\qquad\text{and}\qquad \mathrm{mad}(G[B])<c_2.$$

This is recalled in the source as a main open problem concerning partitions that decrease maximum average degree. The preceding corollaries provide special cases and applications, but the general assertion remains open.