The big problem with contingent

They’re very risky for the applicant. Even if you don't resign from your job before the reference check, you will have to tell your boss you are on the verge of leaving. So either way, you’d be in a vulnerable position with your security in your current job at risk before your offer is final.

As a candidate, one option is to push back on the contingency, explain it could jeopardize your current job, and offer up plentiful other references.