Erdős Problem #19 — Chromatic number of a union of $n$ edge-disjoint copies of $K_n$

Conjecture of Faber, Lovász and myself. Let $G_1, \ldots, G_n$ be $n$ edge-disjoint complete graphs on $n$ vertices. We conjectured more than 20 years ago that the chromatic number of $\bigcup_{i=1}^{n} G_i$ is $n$. I offer 500 dollars for a proof or disproof.