The Donkin–Koppinen filtration conjecture for the polynomial algebra

Let $R_{pol}$ be the algebra of polynomial invariants, with generators $C_r$, $\sigma_i(C_{00})$, $\sigma_j(C_{11})$, $\sigma_n(C_{11})$, and $Ber(C)$ as in the preceding construction. Donkin–Koppinen filtration conjecture. The algebra $R_{pol}$ is generated by the elements

$$C_r, \sigma_i(C_{00})^p, \sigma_j(C_{11})^p, \sigma_n(C_{11})^p Ber(C)^k$$

for $r\geq 0$, $1\leq i\leq m$, $1\leq j\leq n$, and $0<k<p$. This conjecture gives an explicit generating set for the polynomial invariant algebra and is the conjectural step used to obtain the stated filtration result.