On tensoring with the Steinberg representation

Let $G$ be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic $p>0$. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of Donkin: one on tilting modules and the lifting of projective modules for Frobenius kernels of $G$ and another on the existence of certain filtrations of $G$-modules. A key question related to these conjectures is whether the tensor product of the $r$th Steinberg module with a simple module with $p^{r}$th restricted highest weight admits a good filtration. In this paper we verify this statement when (i) $p\geq 2h-4$ ($h$ is the Coxeter number), (ii) for all rank two groups, (iii) for $p\geq 3$ when the simple module corresponds to a fundamental weight and (iv) for a number of cases when the rank is less than or equal to five.