This is a question about the second point in Geordie Willamson's answer in

What to do now that Lusztig's and James' conjectures have been shown to be false? ,

which says that the Lusztig conjecture doesn't need $p\geq h$. I only have heard of the conjecture and proof about the regular block, which requires $p\geq h$. Of course, if there exists a regular block, translating shows that a singular block has character formula with parabolic KL polynomials. But how do you get the result for the $p<h$ case? Does it use the KL correspondence to the affine Lie algebra? Is there any place I can see this explained?