Symmetric weak nilpotence conjecture for the identities $t_n=0$

Let $t_i=0$ denote the corresponding identities for the symmetrized product, and let weak nilpotence of index $n$ have its usual meaning for that product.

Symmetric weak nilpotence conjecture. The identity $t_4=0$ does not imply $t_5=0$, and hence does not imply the index-$4$ weak nil property; the identities $t_4=t_5=0$ imply the weak nil property of index $4$; and, for each $n$, the identities $t_n=t_{n+1}=\cdots=t_{2n-3}=0$ imply $t_{2n-2}=0$, and hence imply weak nilpotence of index $n$.

The conjecture is presented as an analogue of the noncommutative case. The preceding proposition establishes related positive results, including that $t_3=0$ implies weak nilpotence of index $3$, but the three asserted claims remain open here.