Ternary multiplication nilpotence conjecture under $w_7^{(3)}=0$

Let $A$ be a ternary algebra satisfying the identity $w_7^{(3)}=0$. Let $L(A)$ be the associative algebra generated by left multiplications $l_{a,b}\colon x\mapsto (a b x)$, and let $R(A)$ be the associative algebra generated by right multiplications $r_{a,b}\colon x\mapsto (x a b)$.

Ternary multiplication nilpotence conjecture. The associative algebras $L(A)$ and $R(A)$ are nilpotent; in both cases, the nilpotence is of index $13$ and is not less in general.

The claim is presented as a speculation based on the known binary case. The asserted index and sharpness remain open.