Donkin's filtration conjecture

Let $G$ be a semisimple algebraic group over $k$, let $r\geq 1$, let $M$ be a rational $G$-module, and let $\operatorname{St}_r$ denote the $r$-th Steinberg representation. A good $(p,r)$-filtration is a filtration whose factors are of the form $L(\lambda)\otimes\nabla(\sigma)^{(r)}$ with $\lambda\in X_r$ and $\sigma\in X^+$. Donkin's filtration conjecture. For the assertions stated in the source, (a) if $M$ has a good $(p,r)$-filtration, then $\operatorname{St}_r\otimes M$ has a good filtration, equivalently $\operatorname{St}_r\otimes L(\lambda)$ is tilting for $\lambda\in X_r$; (b) if $\operatorname{St}_r\otimes M$ has a good filtration, then $M$ has a good $(p,r)$-filtration; and (c) $M$ has a good $(p,r)$-filtration if and only if $\operatorname{St}_r\otimes M$ has a good filtration. The conjecture links good $(p,r)$-filtrations with tensoring by the Steinberg representation and encompasses Jantzen's question. Its resolution status is not specified in the supplied material.