Nilpotence-index-15 conjecture for multiplication algebras under $w_5=0$

Let $A$ be an algebra satisfying the identity $w_5=0$. Let $L(A)$ and $R(A)$ be the associative algebras generated, respectively, by left and right multiplications.

Nilpotence-index-15 conjecture. The associative algebras $L(A)$ and $R(A)$ are nilpotent; in both cases, the nilpotence is of index $15$ and is not less in general.

The claim was checked once by computer, but the computation took several days and had not received independent verification. Thus the asserted index and its sharpness remain open.