Infinitude of composite numbers of the form $n\# \pm 1$

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Are there infinitely many composite numbers of the form

$$n\# + 1$$

where $n$ is a prime number? What about $n\# - 1$?

Here $n\#$ denotes the primorial function of $n$, i.e. the product of all primes less than or equal to $n$.

I know the proof for the factorial version $n! \pm 1$, but I have no idea where to start from for this one. Any help will be appreciated!