His controversial single-payer health plan has already come up in two races.
Michael Murphy said he wouldn't run for governor next year.
Olson represented George W. Bush in the 2000 recount and later joined with his opposing counsel to challenge California's gay-marriage ban.
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Cuomo's appearance at the NGA in D.C. provoked 2020 buzz.
Tuesday's contest is being closely watched for its presidential election-year implications.
Several Republicans are lining up to run, but it will be difficult to remove Collins from the ballot.