Geordie made some excellent points, beyond my earlier community-wiki comments, but maybe it's worth spilling more pixels to emphasize another important but often neglected viewpoint: Lusztig's 1980 Generic Decomposition Conjecture (GDC), closely related to what is usually called the Lusztig Conjecture (LC). Here is a brief overview of the essential literature:

In a 1977 paper (J. Algebra 49), Jantzen worked out some ideas about "generic decomposition" of Weyl modules in sufficiently large characteristic $p$. This reveals patterns of composition factors (interchangeable in a sense for different alcove types) for all highest weights in general position in the lowest $p^2$-alcove for an affine Weyl group $W_p$ relative to $p$ of Langlands dual type. (For example, in type $B_2$ he finds generically 20 composition factors with multiplicity 1, whereas for $G_2$ there are 119 composition factors with multiplicity ranging from 1 to 4.) He later proved existence more efficiently in a 1980 Crelle J. 317 paper, also in German.

Jantzen's patterns look rather unpredictable, but a "dual" version (which I observed at first indirectly via the finite groups of Lie type) shows symmetry around a special point for $W_p$. From these generic patterns Jantzen also showed how to derive all patterns for arbitrary weights in the region. But the underlying assumption is always that $p$ is at least the Coxeter number $h$, to ensure existence of weights inside $p$-alcoves and permit use of translation functors. To get generic patterns, $p$ of course has to be even bigger in general.

In his write-up of a survey of problems at the 1979 AMS summer institute in Santa Cruz, published in PSPM (1980), Lusztig first stated LC with the hypothesis $p \geq h$ and the stipulation that the dominant weight involved should lie inside an alcove within the "Jantzen region" in the lowest $p^2$-alcove for $W_p$; then the irreducible character should be a $\mathbb{Z}$-linear combination of Weyl characters with coefficients up to sign given by specializations at 1 of Kazhdan-Lusztig polynomials for the (abstract) affine Weyl group.

Eventually Andersen-Jantzen-Soergel proved this (via the successful quantum group version) for "sufficiently large" $p$, after which Fiebig derived an explicit but huge lower bound on $p$. As Williamson has observed, there are serious problems about smaller $p \geq h$.

In 1980 a more technical paper by Lusztig (in Advances 37) laid out a generic version of Jantzen's ideas. Here one has the symmetric dual patterns of alcoves in the abstract affine Weyl group, with polynomial entries in $q$ given by inverse K-L polynomials. His Remark 1.9 is the conjecture that for $q=1$ one should recover Jantzen's patterns, for $p$ "sufficiently large".

In a 1985 paper, S.I. Kato (Advances 60) calls this the GDC and proves in Thm. 5.6 that for "sufficiently large" $p$ it is equivalent to the LC. He builds on a previously accepted but later published 1986 paper by Andersen (*Advances 60) which studies inverse K-L polynomials in more detail.

With this background, it makes sense to distinguish three ranges for $p$: always we need $p \geq h$ (which is a serious limitation for type $A_n$ and related symmetric group problems), whereas for "very large" $p$ both LC and GDC are simultaneously true. For a range of "intermediate" $p$ and for $p<h$ there is serious uncertainty about what if anything to conjecture. Taking GDC as the main focus, I suggest that one try to bound $p$ below as efficiently as possible so that all of Janzten's generic decomposition patterns around some special point lie inside the Jantzen region of the dominant Weyl chamber. In view of the equivalence of GDC and LC for "sufficiently large" $p$, and Verma's older conjecture that the essential data should be independent of $p$, this might lead to a better lower bound on $p$ based on root system constants such as the Weyl group order $|W|$. (Unless this conflicts with Williamson's estimates.)

It's clear that the largest generic pattern will involve at least $|W|$ copies of the "box" (Lusztig's term) at the special point, while the box itself contains $|W|/f$ of the $p$-alcoves with $f$ the index of the root lattice in the weight lattice. But typically there will be more $p$-alcoves involved, containing non-restricted weights below restricted ones (as in $G_2$ etc.). Since the number of $p$-alcoves in the lowest $p^2$-alcove is only $p^\ell$ (with $\ell$ the rank), we are already forced to make $p$ rather large in order to get GDC.