Sites 1-3 will still gossip (well, probably, we already know about convergence, because they know that 4 matters.) As they continue to gossip, they will exchange notes about who saw this value.
Node 4 (and 3 indeed), and all subsequent nodes that are aware of convergence, continue to gossip, but without the random weighting mentioned in your other question. The protocol section of your linked document says:
, gossip , , node. ( ), .