Exact diameters of the twisted wreath counterexample family

Determine the diameters of the Saxl graphs arising from the primitive twisted wreath type groups constructed in Theorem 4.1, beyond the established lower bound of three.

Background

The twisted wreath construction produces an infinite family of primitive groups with base size two whose Saxl graphs contain two nonadjacent vertices with no common neighbour. Therefore, each such graph has diameter at least three.

Unlike the affine and product type constructions, the paper does not establish an upper bound for these graphs. The authors explain that the three-step argument used elsewhere cannot apply because a regular orbital has insufficient out-valency, and explicitly state that the diameters remain unknown.