Babelon–Bernard–Smirnov's surjectivity conjecture for the KdV evaluation map

Let ${\cal D}$ be the ring of commuting KdV derivations, let $A$ be the space of KdV fields, and let $\bar S_{2n}$ be defined by

$$\exp\left(-\sum_{k=1}^{\infty}\frac{1}{k}J_{2k}z^{-2k}\right)=\sum_{n=0}^{\infty}\bar S_{2n}z^{-2n}.$$

Using the identification $H_{-1}^\ast\simeq {\mathbb C}[\bar S_2,\bar S_4,\ldots]$, define the evaluation map

$$\operatorname{ev}_1:{\cal D}\otimes {\mathbb C}[\bar S_2,\bar S_4,\ldots]\longrightarrow A$$

by $P(\partial)\otimes\bar S_2^{\alpha_2}\bar S_4^{\alpha_4}\cdots\mapsto P(\partial)(S_2^{\alpha_2}S_4^{\alpha_4}\cdots)$. Babelon–Bernard–Smirnov's surjectivity conjecture. The map $\operatorname{ev}_1$ is surjective. The conjecture asserts that every KdV field is obtained from the even KdV integrals by applying commuting KdV derivations. The supplied text does not establish its status.