Conjecture on null pairs in universal Lie nilpotent associative algebras

Let $A$ be an associative algebra, and let $L_1(A)=A$ and $L_{k+1}(A)=[A,L_k(A)]$ denote its lower central series. A pair $(i,j)$ is called null when, for every associative algebra $A$, the product $L_i(A)L_j(A)$ is contained in $M_{i+j-1}(A)$, where $M_k(A)=A L_k(A) A$. Null-pair conjecture. A pair $(i,j)$ is null if and only if $i$ or $j$ is odd. The conjecture predicts a complete characterization of the pairs for which the corresponding lower-central-series product satisfies this universal containment. The claim is open here; the paper reports computational evidence that $(2,6)$ and $(4,4)$ are not null, complementing the verified cases for pairs with sum at most $7$.