The 7-Engel conjecture for the identity $t_4=0$

Let $A$ be an algebra with the symmetrized-product identity $t_4=0$. The $m$-Engel identity is the corresponding iterated multiplication identity of length $m$.

7-Engel conjecture. The identity $t_4=0$ implies the $7$-Engel identity.

This is motivated by the known implication from $t_3=0$ to the $3$-Engel identity. The analogous implication for $t_4=0$ is left as a conjecture.