Alexandersson–Shapiro square-graph cone conjecture for derivative discriminants

Let $P(t)=\prod_{i=1}^n(t-x_i)$ be a monic polynomial of degree $n$, and for $0\le k\le n-2$ define

$$\mathcal D_k^{(n)}:=\operatorname{disc}(P^{(k)}).$$

A symmetrized graph monomial is obtained by symmetrizing a product of root differences over labelled graphs. The square-graph cone is the convex cone generated by such monomials whose edges all have even multiplicity.

Square-graph cone conjecture. For every $n\ge2$ and every $0\le k\le n-2$, the polynomial $\mathcal D_k^{(n)}$ belongs to the square-graph cone. Equivalently, $\mathcal D_k^{(n)}$ is a positive linear combination of symmetric sums of products

$$\prod_{(i,j)\in E(G)}(x_i-x_j)^2.$$

The conjecture strengthens a question of Sottile and Mukhin about expressing these symmetric, translation-invariant discriminants through nonnegative expressions in pairwise root differences. The terminal cubic family $k=n-3$ is proved in the paper, while the general square-graph positivity problem remains open.