Least order of a nonstandard finite graph locally K(7,3)

Determine the least order of a finite graph locally isomorphic to K(7,3) other than the standard 120-vertex graph K(10,3).

Background

The paper constructs a 240-vertex graph whose every open neighborhood is isomorphic to K(7,3), thereby providing a nonstandard example beyond the standard 120-vertex K(10,3). The construction is a connected binary double cover of K(10,3), so every graph obtained through the paper’s binary-cover mechanism has 240 vertices.

The authors explicitly state that their construction does not settle the broader extremal question of how small a nonstandard finite graph locally K(7,3) can be. In particular, the cohomological classification of binary covers over K(10,3) supplies no lower bound for finite locally K(7,3) graphs arising by other mechanisms.