The half-factorial residue distribution conjecture for primes congruent to 3 modulo 4

Let $p$ be a prime with $p \equiv 3 \pmod 4$. Consider the proportion, among such primes, for which

$$\left(\frac{p-1}{2}\right)! \equiv 1 \pmod p.$$

Half-factorial residue distribution conjecture. This proportion approaches $\frac{1}{2}$ as $p \to \infty$.

Numerical evidence motivates the conjecture. It is related to the parity of the number of quadratic non-residues below $p/2$ and has been recast in terms of the class number of $\mathbb Q(\sqrt{-p})$, but no proof is known.