Generalized Knight conjecture for first-order sentences in random groups

Let $d<1/16$, and let $\Gamma_\ell$ be a random group of density $d$ at length $\ell$. A first-order sentence is almost surely true in this model if the probability that it holds in $\Gamma_\ell$ tends to $1$ as $\ell\to\infty$. Generalized Knight conjecture. Suppose $\sigma$ is a first-order sentence in the language of groups. Then $\sigma$ is almost surely true in the random group of density $d$ if and only if it is true in a nonabelian free group. The paper proves the corresponding assertion for universal sentences, while this extension to all first-order sentences is left as a conjecture.