by conflict, I mean that the turn itself logically conflicts with pedestrians/bicyclists/perpendicular vehicle traffic going across the intersection on their green light, and potentially also people in the parallel crosswalk finishing their movement.

The statute is here: https://law.justia.com/codes/new-jersey/2021/title-39/section-39-4-115/

Turning right (on green or red) requires that:

1. the driver intends to turn
2. the intersection is controlled by a signal or officer
3. proper care is taken to prohibit accidents by the driver

Additionally, if the light is red, it is required for the driver to:

4. not turn if there is a sign disallowing a turn on red
5. come to a full stop
6. observe traffic in ALL directions
7. yield to vehicles traveling in the direction of the turn
8. stop and remaining stopped for pedestrians crossing the roadway in a marked or unmarked crosswalk
9. make the turn from and to the rightmost part of both roads unless posted otherwise.
10. Only then shall they proceed to make a turn.

First, a driver can choose to not intend to turn right until there's a green light. Second, a driver is required to take proper care to prevent accidents and observe traffic in all directions, yielding to vehicles and pedestrians. Turning right on red is inherently more dangerous than proceeding on a green signal. They might observe traffic and take proper care longer than you'd do it, and might consume the entire red light depending on their judgment, and that's ok.

Both of these happen in time before the "shall proceed" section.