Sharpness conjecture for products of commutators in associative rings

Let $m,n\in\mathbb Z$ satisfy $m,n>1$, and suppose that at least one of $m$ or $n$ is odd. Let $T^{(k)}$ denote the relevant subgroup or ideal generated by products of $k$-fold commutators in a unital associative ring.

Sharpness conjecture. If $(m,n)\ne(3,3)$, then there is a unital associative ring $A$ and elements $a_i,b_j\in A$ such that

$$[a_1,\dots,a_m][b_1,\dots,b_n]\notin T^{(m+n-1)}.$$

The claim asserts that, apart from the exceptional case $(m,n)=(3,3)$, the coefficient $3$ in the preceding theorem cannot generally be removed. The cited discussion gives evidence for this sharpness, while the $(3,3)$ case is known to behave exceptionally.