The twisted-homogeneous-variety tilting conjecture

Let $X$ be a twisted form of a homogeneous variety, meaning a variety that becomes homogeneous after a suitable field extension. A tilting object in $D^b\operatorname{Coh}(X)$ is an object with no higher self-Ext groups that generates the derived category.

Twisted-homogeneous-variety tilting conjecture. The category $D^b\operatorname{Coh}(X)$ admits a tilting object.

The conjecture is motivated by tilting objects on generalized Brauer--Severi varieties and by the fact that twisted forms need not admit full strongly exceptional collections. The source proves the assertion only for certain twisted forms, leaving the general claim open.