Laziest connected graph with connected complement

Determine whether the double star DS(n−3, 1) is the laziest connected graph on n vertices whose complement is connected.

Background

The paper studies the laziness of a graph, defined as the trace of the average mixing matrix associated with its Laplacian quantum walk. It proves that the complete graph is the laziest connected graph, that the star is the second laziest connected graph, and that the double star DS(n−3, 1) is the second laziest tree. Computational data for graphs with up to nine vertices motivates asking whether this extremal result extends to the narrower class of connected graphs whose complements are also connected.