Degree–singularity–root correspondence for nilpotent Toda lattices

Let $p(q)$ be the polynomial associated with the relevant $\Gamma_{-}$-polytope, let $\eta(w_*)$ denote its degree, and let $p_o$ be the point where all divisors ${\mathcal D}_j=\{\tau_j=0\}$ intersect. Set ${\mathcal D}_0=\bigcup_{j=1}^l{\mathcal D}_j$, and let $d$ be the multiplicity of the singularity of ${\mathcal D}_0$ at $p_o$.

Degree–singularity–root conjecture. The degree $\eta(w_*)$ of $p(q)$ is the multiplicity $d$ at $p_o$, where $d$ is given by the minimal degree of the product of the $\tau$-functions; moreover, $\eta(w_*)$ is equal to the number of real roots of the Schur polynomials associated with the nilpotent Toda lattices.

The claim connects the polynomial degree with the intersection singularity of the Toda divisors and with real-root counts for nilpotent Toda flows. The supplied text presents it in a Conjecture environment but gives no resolution evidence; the attached citation is retained as external context.