Prime factorization conjecture for factorials plus one

Let $S=\{4,5,7,12,23,229,562\}$. A natural number is square-free if no perfect square other than $1$ divides it.

Prime factorization conjecture for factorials plus one. For every $n\in\mathbb N\setminus S$, the number $n!+1$ is square-free.

The conjecture predicts that the displayed set contains all exceptions to square-freeness for numbers of the form $n!+1$. Numerical evidence is reported through $n=100$, while the further exceptions $n=229$ and $n=562$ are explicitly identified in the source; no proof that these are the only exceptions is given.