Equal color-weight 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 let $a_i$ and $b_i$ denote the quantities associated with color $i\in[r]$. Equal color-weight conjecture. For every $i,j\in[r]$, one has

$$a_i+b_i=a_j+b_j.$$

The paper describes this as a strengthening of the preceding proposition and reports that the authors were unable to prove it; its status therefore remains open.