The invariant ring $S(\mathfrak{h})^W$ in bad characteristic

Question

Let $G$ be a semisimple algebraic group (with arbitrary Dynkin diagram for now) of adjoint type defined over an algebraically closed field $K$ of not very good characteristic $p$. Then the action of the Weyl group $W$ on a Cartan subalgebra $\mathfrak{h}$ is not necessarily reducible (as opposed to the characteristic 0 case). Is it known in which cases the invariant ring $S(\mathfrak{h})^W$ is polynomial? A direct computation says that this is the case for $G = PGL_2, p = 2$, since then the $W$-action on $\mathfrak{h}$ is trivial. I'm interested in whether there has been any progress made on this question in the literature. It seems feasible to compute this directly for $G = PGL_{n+1}$, $p|n+1$ (perhaps for small $n$), but some of the other types are going to be more difficult.