This is half-way between a comment and an answer:
Lusztig's quantum group conjecture in Section 8 of this paper:
Lusztig, G, Modular representations and quantum groups. Classical groups and related topics (Beijing, 1987), 59–77, Contemp. Math., 82, Amer. Math. Soc., Providence, RI, 1989.
Note how clean the statement is: no $p > h$, no restriction on highest weight etc. This is what meant when I say that this is the perfect conjecture. On the other hand I didn't say that we have a perfect proof!
I don't understand precisely what is known and where it is written, and would prefer not to enter into this maze.
Let me outline why this should be true. In the recent preprint with Riche
we explain that if one has an action of the Hecke category on the principal block then one gets a complete description of the category "for free". (I.e. getting the action might be hard, but things are easy once one has it. Roughly one can think that the principal block is a cyclic module under the Hecke category.)
Note: The above paper is concerned with the algebraic group. However it is easy to run the same arguments for the easier case of the quantum group in which case all occurrences of the p-canonical basis are replaced by the Kazhdan-Lusztig basis.
There is a singular version of this conjecture: basically if one groups all blocks together one should have an action of singular Soergel bimodules on everything (the singularity is determined by the block, see Conjecture 1.6 in the above paper for an idea). This now makes sense when there is no regular block. Similarly everything is controlled by the Hecke category, and character formulas follow.