Usually the question whether the [diamond principle][1] $\diamondsuit(\kappa)$ holds for some large cardinal $\kappa$ only concerns large cardinal notions of very low consistency (among the weakly compacts). Partly since it *does* hold for all [subtle cardinals][2], which are only barely stronger than the weakly compacts, and pretty much every large cardinal notion below a weakly compact has been shown to consistently *not* satisfy it (see https://mathoverflow.net/questions/137036/failure-of-diamond-at-large-cardinals and [Ben Neria ('17)][3]).

That subtle cardinals satisfy diamond of course means that almost all large cardinals *do* satisfy it as well, but there are some strange ones lying around though, including [Woodin cardinals][4] and inaccessible [Jónsson cardinals][5]. Is anything known about diamond holding for any of these two?


  [1]: https://en.wikipedia.org/wiki/Diamond_principle
  [2]: http://cantorsattic.info/Ineffable#Subtle_cardinal
  [3]: https://arxiv.org/abs/1705.01611
  [4]: https://en.wikipedia.org/wiki/Woodin_cardinal
  [5]: http://cantorsattic.info/Jonsson