Projected-stationarity conjecture for complete Das–Dennis grids

Let $\Delta_2$ be the two-dimensional simplex, let $G_H$ be the complete Das–Dennis grid for $H\geq 2$, let $U_H$ denote the corresponding union of anchored boxes, and let $D(S)$ be the dominated region generated by a set $S$. For a point $p$ and feasible tangent direction $d$, consider replacing $p$ by $p+\varepsilon d$ in the grid.

Projected-stationarity conjecture. For every $H\geq 2$, the complete Das–Dennis grid $G_H$ is projected-stationary for anchored-box magnitude on $\Delta_2$. More precisely:

1. If $p\in G_H$ is not a simplex vertex and $d$ is any feasible tangent direction through $p$, then

$$\operatorname{Mag}\bigl(D((G_H\setminus\{p\})\cup\{p+\varepsilon d\})\bigr)=\operatorname{Mag}(U_H)-c_{p,d}\,\varepsilon^2+o(\varepsilon^2)$$

for some $c_{p,d}\geq 0$.

2. If $v\in G_H$ is a simplex vertex and $d$ is any feasible inward direction, then

$$\operatorname{Mag}\bigl(D((G_H\setminus\{v\})\cup\{v+\varepsilon d\})\bigr)=\operatorname{Mag}(U_H)-\gamma_{v,d}\,\varepsilon+o(\varepsilon)$$

with $\gamma_{v,d}>0$.

The conjecture formalizes the numerical evidence that complete Das–Dennis grids are stationary under feasible perturbations: non-vertex points have no first-order variation and a nonpositive quadratic term, whereas simplex vertices satisfy a strict one-sided first-order decrease. A proof would explain the observed local cancellation of anchored-box contributions, but the conjecture remains open based on the supplied source.