All-two tuple conjecture for maximal lower-central-series containments

Let $A_n$ be the free associative algebra on $n$ generators, and let $j=(2,\dots,2)\in\mathbb{N}^k$. All-two tuple conjecture.

$$\mathrm{I}(A_n,j)= \begin{cases} \max\left\{2\left\lceil\frac{k}{2}-\frac{n}{4}\right\rceil,0\right\}+2 & n\geq4\text{ and }n\text{ even},\\ \max\left\{2\left\lceil\frac{k}{2}-\frac{n}{4}+\frac{1}{4}\right\rceil,0\right\}+2 & n\geq5\text{ and }n\text{ odd},\\ k+1 & \text{otherwise.} \end{cases}$$

The formula is motivated by combinatorial considerations and was checked in small cases; its general validity remains open.