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- Primes P such that ((P-1)/2)!=1 mod P 5 answers

I came across the following question:

Is it known whether there exist infinitely many primes $p$ such that $\frac{p-1}{2}!\equiv_{p}1$?

(It is easy to see that if $p\equiv_{4}3$ then this number is $\pm 1$ mod $p$.)