It is a well-known conjecture that there exist infinitely many primes of the form $n!+1$.

This problem is somewhat similar to the famous Landau's problem about the infinitude of the primes of the form $n^2+1$. Iwaniec has shown that $n^2+1$ is a product of at most two primes for infinitely many $n$. Is there a similar result regarding $n!+1$?

For example, is there any partial progress towards this problem which places some bound on $\omega(n!+1)$ for infinitely many $n$?