What is the conjecture

https://www.erdosproblems.com/1074

Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m!+1\equiv 0\pmod{p}$. Does
$$\lim \frac{\lvert S\cap [1,x]\rvert}{x}$$
exist? What is it?

Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1\pmod{m}$ such that $m!+1\equiv 0\pmod{p}$, then does
$$\lim \frac{\lvert P\cap [1,x]\rvert}{\pi(x)}$$
exist? What is it?

Status: open

Choose either option

• I plan on working on this conjecture
• This issue is up for grabs: I would like to see this conjecture added by somebody else