Várilly-Alvarado and Vakil's density-function conjecture for scrollar invariants

For $x\in\mathbb{R}^{d-1}$, let $\{B_i(x)\}$ be open balls containing $x$ whose radii converge to zero, and define

$$\rho^{\textnormal{geo}}(x)=\lim_{i\to\infty}\pi^{\textnormal{geo}}(B_i(x)),$$

where the limit is taken whenever the geometric limiting measure exists. Define $\rho^{\textnormal{gr}}$ analogously using $\pi^{\textnormal{gr}}$.

Várilly-Alvarado and Vakil's density-function conjecture. The quantities $\rho^{\textnormal{geo}}(x)$ and $\rho^{\textnormal{gr}}(x)$ exist and are independent of the choice of $\{B_i(x)\}$. They satisfy

$$\rho^{\textnormal{geo}}=\dim\circ\rho^{\textnormal{gr}}.$$

Moreover, they are supported on a finite union of rational polytopes, are piecewise linear, and are continuous on their support. This conjecture concerns the local limiting density of scrollar-invariant strata and remains open in the setting discussed by the paper.