See the TV series The Americans for a fictional depiction.

You know product ABC has ongoing yield problems from an outside source. Wait for moment when Person Z looks burnt out/stressed/etc and walk through emotional available "you can talk to me about anything" routine.

Don't pry about specific topic but wait until Person Z volunteers some vague conflict at work you know is the yield issue and try to gently redirect conversation using your insider knowledge Person Z isn't aware of to specific area you're interested in.