The rank formula conjecture for the third lower central series quotient

Let $A=\mathbb{Z}\langle x,y\rangle/(x^m+y^m)$, and let $N_3(A)$ denote the third lower central series quotient, with $N_3(A)[d]$ its degree-$d$ component. Rank formula conjecture. For every degree $d$, one has

$$\operatorname{Rank}(N_3(A)[d])=\begin{cases}3d-7 & 3\leq d\leq m+1 \\ 6m-3d+1 & m+2\leq d\leq 2m \\ 0 & \operatorname{elsewhere}. \end{cases}$$

This conjecture is suggested by computations of the higher lower central series quotients, specifically the case $k=3$, and predicts the complete graded rank of $N_3(A)$. Its resolution would clarify the structure of lower central series quotients for finitely generated algebras over $\mathbb{Z}$.