The density conjecture for prime divisors absent from the $r$-derangement numbers

For a fixed positive integer $r$, let $C_r(n)$ denote the $r$-derangement numbers, and let $\mathbb{P}$ be the set of prime numbers. Define

$$\mathcal{A}_r:=\{p\in\mathbb{P}:\forall_{n\geq r}\ p\nmid C_r(n)\}.$$

Density conjecture. The set $\mathcal{A}_r$ is infinite. Moreover,

$$\lim_{n\rightarrow +\infty} \frac{\sharp (\mathcal{A}_r\cap\{1,...,n\})}{\sharp (\mathbb{P}\cap\{1,...,n\})} = \frac{1}{e}.$$

The preceding result establishes that the complementary set of primes dividing at least one number $C_r(n)$ is infinite. The asserted infinitude and natural density $1/e$ of the primes that divide none of the $r$-derangement numbers remain unproved in the supplied text.