The perfect diameter-two graph conjecture for 2-bootstrap percolation

Let $\mathcal{G}$ be the collection of all connected graphs of order at least $3$ with at most two blocks. A graph is 2-bootstrap good if it contains at least one pair of vertices that percolate under the $r=2$ bootstrap percolation process. A graph is perfect if every induced subgraph has a vertex whose neighborhood is a clique, and its diameter is the maximum graph distance between two vertices. The perfect diameter-two graph conjecture. If a graph in $\mathcal{G}$ is perfect and its diameter is no more than $2$, then the graph is 2-bootstrap good. This would give a sufficient structural condition for 2-bootstrap goodness within the class $\mathcal{G}$; the supplied text does not indicate whether the conjecture has been resolved.