Conjecture on decay of mixed derivatives of the symmetric coordinates

Let $u_r$ be the symmetric homogeneous polynomial coordinates in $N$ variables constructed in the main theorem. For a partition $\sigma\vdash r$ and a differential operator of order $r$ corresponding to $\sigma$, write $l(\sigma)$ for the length of the partition. Then decay conjecture. If $\prod_{q=1}^{r}\partial_{i_q}$ is such a differential operator, the total expression of

$$\frac{\partial^d u_r}{\prod_{q=1}^{d}\partial_{i_q}}$$

decays like $O\bigl(N^{-(r+l(\sigma)-1)}\bigr)$ as $N\to\infty$. This conjecture refines the observed fact that increasingly mixed derivatives decay increasingly rapidly with the number of variables; it is based on explicit calculations for all $r\leq 6$ and remains open in the supplied source.