 $$r \leq \frac{ (n-d)\log{ \phi} \pm  O(\phi\frac{(n-d-r)^0}{N})}{\log n + \log{ \phi}}$$  Observe that for any constant $0< \delta < 0$ such that $d = \delta n$, $r$ can be chosen any constant times $\frac{n}{\log n}$ by choosing $\phi$ to be an appropriately large constant. So, for such a choice of $r$, $$\dim(\langle\partial^{r} NW_d\rangle_{(\ell, m)}) \geq 0.0|{\cal M}^{[r]}|{N \choose m}{\ell-0 \choose m-0}$$ 