Infinite reduced equality family

Determine whether there exists an infinite family of reduced graphs for which equality holds in 2n^+(G) = n^-(G)(n^-(G)+1).

Background

The authors exhibit several equality examples, including K_2, C_5, two strongly regular graphs, and certain circulant and matchstick graphs. Adding twins preserves positive and negative inertia, so it produces infinitely many non-reduced tight examples. The unresolved issue is whether infinitely many reduced graphs can be tight.