I don’t think Cook County rats crossover in IL GOP primaries much at all in any election, there are always contested rat primaries. And they’d have to verbally ask for a GOP ballot and risk getting a dirty look.
In the 2006 primary, when I lived in North Chicago, the election judge didn’t ask which ballot I wanted. About 90% of the voters, in that precinct, are black, and the judge must have assumed that I wanted a democrat ballot. I asked for a republican ballot, and she gave me a democrat ballot. I gave it back and said, “I asked for a republican ballot.” She said, “I don’t know if we got any republican ballots.” She looked for about a minute, and she found one.