The random-group theory conjecture for densities below one-half

Let $0\leq d<\frac{1}{2}$, and let $\psi$ be a first-order sentence in the language of groups. Let $F_k$ be the free group of rank $k\geq 2$, and consider the density-$d$ random-group model on a fixed basis of $F_k$. Random-group theory conjecture. The random group of density $d$ satisfies $\psi$ with overwhelming probability if and only if the free group $F_k$ satisfies $\psi$. This extends the conjectures of Knight and of Kharlampovich and Sklinos relating first-order theories of free groups to those of random groups. The statement concerns all densities below the hyperbolicity threshold $\frac{1}{2}$; the supplied source does not state whether it has been proved or disproved.