Intersection-support conjecture for optimal colorings

For $r\in\mathbb{Z}^{+}$, let $a,b:\mathcal{P}([r])\to[0,1]$ be an optimal solution to the coloring problem, and write $a(R)$ and $b(R)$ for the corresponding aggregate quantities associated with $R\subseteq[r]$. Intersection-support conjecture. There do not exist subsets $R_1,R_2\subseteq[r]$ such that $|R_1\cap R_2|\geq 2$, $a(R_1)>0$, and $b(R_2)>0$. This is a structural conjecture about every optimal solution and is presented without a resolution in the paper.