Quest for D-Affinity

I am interested in the following problem: "classify all smooth connected D-affine projective varieties".
As a part of their proof of Kazhdan-Lusztig Conjecture, Beilinson and Bernstein have proved that
the partial flag varieties G/P are D-affine. This is the current state of art: no other examples are known
but there is no proof that the G/P-s exhaust all the possible examples.

In my talk I will show that in three natural classes of varieties D-affinity implies that the variety is G/P,
review some examples of more general D-affine spaces, and try to convince the listeners that the problem
is related to some other interesting questions in Algebraic Geometry.