Does "the Law written on hearts" refer to conscience or regeneration?

In Romans 2:14-15, Paul speaks of law written on hearts:

14 For when Gentiles, who do not have the law, by nature do what the law requires, they are a law to themselves, even though they do not have the law. 15 They show that the work of the law is written on their hearts, while their conscience also bears witness, and their conflicting thoughts accuse or even excuse them …

In 2 Corinthians 3:2-3, Paul uses this term to refer to believers, seemingly referring to the work of the Spirit in regeneration:

Discussion