The classification conjecture for tuples with the Khintchine property

Let $p$ be a prime, let $k<p$, and let $c_0,\ldots,c_k$ be distinct elements of $\mathbb{F}_p$. A tuple has the Khintchine property when it satisfies the recurrence property defined in the paper. Classification conjecture.

1. If $k>3$, then $(c_0,\ldots,c_k)$ does not have the Khintchine property.
2. If $k=3$ and $(c_0,c_1,c_2,c_3)$ does not form a parallelogram, then $(c_0,c_1,c_2,c_3)$ does not have the Khintchine property.

The preceding results establish the Khintchine property for the indicated double-recurrence tuples and for triple-recurrence tuples forming a parallelogram; this conjecture asserts that these are all the possible cases.