Alternative extremal families for propagation-time discrepancy

Determine whether infinite families of graphs other than the family presented in Theorem 4.1 satisfy $pd(G_n)=\lfloor n-2\sqrt n+1\rfloor$.

Background

The paper proves the upper bound pd(G)≤⌊n−2√n+1⌋ for the propagation-time discrepancy between minimum zero forcing sets and constructs an infinite family attaining equality when n=k^2+1 for odd k≥3. The authors ask whether other infinite extremal families exist.