Primitivity of coherent closures of rank-five Neumaier graphs

Determine whether the coherent closure of every Neumaier graph of coherent rank five is imprimitive, and more strongly, whether it must contain a non-trivial parabolic whose classes are independent sets.

Background

The constructed graphs have adjacency matrices in the Bose–Mesner algebra of the four-class association scheme of Holzmann, Kharaghani, and Suda. That association scheme is imprimitive because it has a non-trivial parabolic. The paper asks whether this imprimitive structure is unavoidable for all Neumaier graphs of coherent rank five, including the stronger possibility that their coherent closures necessarily contain a non-trivial parabolic with independent-set classes.