Rupert Murdoch is now pushing to amend the previously established trust to give the bulk of this power to Lachlan.

At stake is the ideological future of Fox News and News Corp: Unlike Lachlan, the other three elder Murdoch siblings — James, Elisabeth, and Prudence — are viewed as more centrist, and could take the family’s media outlets in a more moderate direction. That concern is among those that spurred Rupert Murdoch to pursue this court action, and to shift more control to the child he sees as most likely to carry on his vision.
https://www.vox.com/politics/372145/murdoch-succession-court-case-rupert-lachlan