The difference between the chromatic and the cochromatic number of a random graph

The cochromatic number $\zeta(G)$ of a graph $G$ is the minimum number of colours needed for a vertex colouring where every colour class is either an independent set or a clique. Let $\chi(G)$ denote the usual chromatic number. Around 1991 Erd\H{o}s and Gimbel asked: For the random graph $G \sim G_{n, 1/2}$, does $\chi(G)-\zeta(G) \rightarrow \infty$ whp? Erd\H{o}s offered $100 for a positive and $1,000 for a negative answer. We give a positive answer to this question for roughly 95% of all values $n$.