Classical affine Schur-Weyl kernel conjecture

Let $E[AS_m]$ be the group algebra of the affine symmetric group, let $\chi:E[AS_m]\to\operatorname{End}_{\operatorname{Sym}(\underline{x})}(R^m_n)$ be the specialization at $q=1$, and let $e_{sgn}(T_1,\dots,T_k)$ denote the sign idempotent in $\mathbb C[S_k]\subset\mathbb C[S_m]$. For $h_j=h_j(X_1,\dots,X_{k+1})$, write $h_j$ for the homogeneous symmetric function of degree $j$, and let $e_r$ denote the relevant elementary symmetric functions. Classical affine Schur-Weyl kernel conjecture. The kernel of $\chi$ is generated, for $k=0,\dots,\min(n,m-1)$, by

$$e_{sgn}(T_1,\dots,T_k)(h_{n-k}-e_1h_{n-k-1}+e_2h_{n-k-2}-\dots+(-1)^{n-k}e_{n-k}).$$

This conjecture interpolates between the Cayley-Hamilton theorem at $k=0$ and Schur-Weyl duality at $k=n$ when $m>n$; the full kernel description is not established in the supplied text.