The extreme-coefficient conjecture for ED polynomials of Segre products

Let $X_1\subset\mathbb{P}(V_1)$ and $X_2\subset\mathbb{P}(V_2)$ be projective varieties, let $Q_i\subset\mathbb{P}(V_i)$ denote the isotropic quadrics, and let $Q$ be the corresponding isotropic quadric for the Segre product. Write the ED polynomial as

$$\operatorname{EDpoly}_{(X_1\times X_2)^\vee,t}(\varepsilon^2)=\sum_{i=0}^{N}c_i\varepsilon^{2i}.$$

Assume that $(X_1\times X_2)\cap Q$ is reduced, and let $f$ and $g$ be square-free polynomials whose vanishing loci are

$$\mathcal{V}(f)=(X_1\times X_2)^\vee,$$ $$\mathcal{V}(g)=[(X_1\cap Q_1)\times X_2]^\vee\cup[X_1\times(X_2\cap Q_2)]^\vee.$$

Extreme-coefficient conjecture. The extreme coefficients are

$$c_0=f^2g,\qquad c_N\in\mathbb{R}.$$

The conjecture is motivated by computations in Macaulay2 and, if true, implies stabilization of the ED degree of $X\times\mathbb{P}(V_2)$ as the dimension of $V_2$ increases; its general validity remains open.