4 problems

Maximality conjecture. Any two maximal level-$k$ hypertriangulations of $A$ have the same, and therefore maximum, number of triangles. Equivalently, every maximal level-$k$ hypertr…

Triangle-number invariance conjecture. The number of triangles in a maximum hypertriangulation depends on the given points but not on how these points are triangulated.

Order-$k$ Delaunay optimality conjecture. The hypertriangulation $P_k$ has the maximum number of triangles if and only if $P_k$ is the order-$k$ Delaunay triangulation of $A$.

Uniqueness conjecture. The order-$k$ Delaunay triangulation is the only level-$k$ hypertriangulation with maximally many triangles that has the local angle property.