What steps have been taken so far to solve Brocard's Problem?

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The equation is $$n!+1=m^2$$ where $n$ and $m$ are natural numbers. Brocard's Problem asks whether there are solutions for n other than $4, 5, 7$. The only improvement I have found that people have done toward solving the problem is that if the ABC conjecture is true, then there are finite solutions for $n$. However, this is all I have found, despite having at least one reward for it. What other things do we know about Brocard's Problem?