Do verbs following relative pronouns need to be in the past tense if the sentence begins in the subjunctive?

"I would prefer someone who was a bit more sincere about their beliefs."

vs.

"I would prefer someone who’s a bit more sincere about their beliefs."

The second sounds more correct to me, since "someone who…" seems to be a single direct object.

However, when I replace "was" with another verb:

"I would prefer someone who cared about me."

vs.

"I would prefer someone who cares about me."

Both of these sound correct (though I suspect the first is technically correct, and the second perfectly fine in everyday vernacular).

Any thoughts?