The primeness conjecture for the equivalence relation $\mathbb E_1$

Let $\mathbb E_1$ be the equivalence relation on sequences of reals defined by eventual agreement, and let a Borel equivalence relation be prime if it is Borel-reducible to no product or join structure covered by the paper's definition of primeness. The primeness conjecture for $\mathbb E_1$. $\mathbb E_1$ is a prime equivalence relation. This is presented as a natural candidate for primeness; the surrounding discussion notes that the conjecture would follow from the Kechris–Louveau conjecture together with the primeness of $\mathbb E_1$ relative to orbit equivalence relations.