Katznelson's recurrence conjecture for $\mathbb F_p^\omega$

Let $p$ be a prime and let $\mathbb F_p^\omega$ be the direct sum of countably many copies of $\mathbb F_p$. A set is a set of Bohr recurrence if it intersects every finite-index subgroup of $\mathbb F_p^\omega$, and a set of chromatic recurrence if its Cayley graph has infinite chromatic number.

Katznelson's finite-field conjecture. Every subset of $\mathbb F_p^\omega$ which is a set of Bohr recurrence is a set of chromatic recurrence.

This is the specialization of the general Katznelson recurrence problem to countably infinite-dimensional vector spaces over finite fields. The paper relates it to the preceding uniform-set conjecture.