Is the conjecture about prime numbers true?

Question

Let $p_n$ be the $n$-th prime number. Is it true that if $n$ is sufficiently large then will $$p_1×p_2×p_3×...×p_n+1$$ always be a composite number?

Answer 1

I tried my best to explain why we always get a composite number.

Case 1: if these primes are arrange in ascending order and $p_1$ is 3 than: $$p_1×p_2×p_3×...×p_n+1$$ is always a composite number as product of $n_th$ odd numbers (here primes) will always odd and adding 1 makes it even.

Case 2: if we take $p_n$ as 2 than $$p_1×p_2×p_3×...×p_n+1$$

will never be an even number as ($p_1×p_2×p_3×...×p_n$) will be even and adding 1 makes it odd.