On Donkin's Tilting Module Conjecture I: Lowering the Prime

In this paper the authors provide a complete answer to Donkin’s Tilting Module Conjecture for all rank $2$ semisimple algebraic groups and ${\text{SL}}_4(k)$ where $k$ is an algebraically closed field of characteristic $p>0$. In the process, new techniques are introduced involving the existence of $(p,r)$-filtrations, Lusztig’s character formula, and the $G_r$T-radical series for baby Verma modules.