Binary weak nilpotence conjecture for the identities $w_n=0$ through $w_{2n-3}=0$

Let $A$ be a nonassociative algebra, and let $w_i=0$ denote the corresponding multilinear identities. The algebra is weakly nilpotent of index $n$ when the stated identities force all sufficiently long products in the relevant weak sense.

Binary weak nilpotence conjecture. For each $n$, the identities $w_n=w_{n+1}=\cdots=w_{2n-3}=0$ imply the identity $w_{2n-2}=0$, and hence imply weak nilpotence of index $n$.

This generalizes the exhibited case in which $w_4=w_5=0$ implies weak nilpotence of index $4$.