The correct answer is false.
It is false that the rise of successor states after the fall of Rome led to a decline in Christianity.
The fall of the Roman Empire occurred in September of 476 AD, when Odovacar, a German Prince, deposed Emperor Romulus Augustus. After 500 years of total dominance, the Roman Empire collapsed. Many kings began to rule their different states. But one thing that did not happen was the decline of Christianity, This religion was already ingrained in most of the people that had lived under the Roman Empire rule.