Prove or disprove that $\gcd\left(n!+1,f\left(n\right)\right)=1$

Question

Problem/Conjecture :

Let :

$$f(x)=\sum_{i=1}^{\infty}\left\lfloor\frac{x^{i}}{i!}\right\rfloor$$

Then it seems we have $n\geq 1$ an integer :

$$\gcd\left(n!+1,f\left(n\right)\right)=1$$

This question is an attempt to find a proof to Do we have $\sum_{n=1}^{\infty}\frac{\gcd\left(1+n!,1+n^{2}\right)}{n!}\stackrel{?}{=}e$?

It's true for $1\leq n\leq 22$

For a proof except Wilson's theorem I don't see . On the other hand a couter-example is very very welcome .

Remark :

For $1\leq n\leq 100$ it seems we have as prime factorization :

$$n!+1=p_1^{a_1}p_2^{a_2}\cdots p_r^{a_r}$$

then :

$$p_i\geq n,1\leq i \leq r$$

If the remark is true in the general case the inequality mean it generates larger and larger prime number .

How to (dis)prove it ?

Any things which make progress is very welcome .