It’s not a conflict, it’s a hierarchy of preferences. People’s specific opposition to race-conscious admissions and hiring outweighs their general support for diversity. That’s exactly what the Pew answer choice says: “Only take a person's qualifications into account, even if it results in less diversity.” Race blindness is more important than diversity.
You see a similar pattern on many issues. E.g. people support balancing the federal budget. But most would say that they oppose cutting entitlement spending to balance the budget. That is, people’s specific opposition to cutting entitlement spending outweighs their general support for balancing the budget. In practice it just means that people’s preference in favor of the general issue is weakly held, while their opposition on the specific issue is strongly held.
You see a similar pattern on many issues. E.g. people support balancing the federal budget. But most would say that they oppose cutting entitlement spending to balance the budget. That is, people’s specific opposition to cutting entitlement spending outweighs their general support for balancing the budget. In practice it just means that people’s preference in favor of the general issue is weakly held, while their opposition on the specific issue is strongly held.