Shalev's probabilistic identity conjecture for finitely generated pro-$p$ groups

Let $G$ be a finitely generated pro-$p$ group. A probabilistic identity is a word $w$ such that the probability that random elements satisfy $w$ is positive, and an identity is a word law satisfied by all substitutions from $G$.

Shalev's probabilistic identity conjecture. If $G$ satisfies some probabilistic identity $w$, then $G$ satisfies some identity.

This is one of Shalev's conjectures on the relationship between probabilistic identities, coset identities, and ordinary identities in pro-$p$ groups. Its status is not specified in the supplied text.