The eigenvalue-support conjecture for rational KP solitons

Let $M=(X,Y,Z)\in\mathcal{M}_n^1$ represent a rational KP soliton with normalized Baker–Akhiezer function $\psi_M$. Consider the finitely supported distributions in $z$ that annihilate $z^n\psi_M$; their support records the points at which these distributions are concentrated, and the Jordan blocks of $Y$ and $Z$ determine the corresponding derivative orders.

Eigenvalue-support conjecture. The support of the distributions annihilating $z^n\psi_M$ is the set of eigenvalues of the matrices $Y$ and $Z$, with the highest derivative taken at a particular eigenvalue bounded by the size of the corresponding Jordan block.

For non-degenerate solitons, the eigenvalues of $Y$ and $Z$ already determine the support and $X$ determines the coefficients. This conjecture asserts that the same description remains valid in general, clarifying the geometry of $\mathcal{M}_n^1$ and its relationship with the rational Grassmannian and KP solutions.