The even-index lower central series product noncontainment conjecture

Let $A$ be an associative algebra with lower central series ideals $M_i$.

Even-index product conjecture. If $i,j\geq 2$ are both even, then in general

$$M_i\cdot M_j\nsubseteq M_{i+j-1}.$$

This contrasts with the preceding result that $M_iM_j\subset M_{i+j-1}$ when at least one of $i,j$ is odd and $\frac{1}{6}\in R$; the statement asserts that the analogous containment generally fails when both indices are even.