Dummit–Ford–Kisilevsky–Sands conjecture for Gauss factorials

Let $G_n$ denote the Gauss factorial, and call a prime $p$ 1-exceptional for $m=3$ when the corresponding exceptional congruence holds. Dummit–Ford–Kisilevsky–Sands conjecture. There are infinitely many primes $p\equiv 1\pmod 3$ such that

$$G_{p-1}\equiv 0\pmod {p^2}.$$

Equivalently, there are infinitely many primes $p\equiv 1\pmod 3$ such that $p$ is 1-exceptional for $m=3$. This is the specialization to $K=\mathbb{Q}(\sqrt{-3})$ of the conjecture that, for every fixed imaginary quadratic field $K$, infinitely many primes satisfy $\lambda_p(K)>1$; its status is open.