Terminal stability conjecture for derivative discriminants

For $n\ge r$, let $\mathcal D_{n-r}^{(n)}=\operatorname{disc}(P^{(n-r)})$, where $P$ is a monic polynomial of degree $n$. For fixed $r$, let $G_1,\ldots,G_N$ be square multigraphs and let $\widetilde{G_\nu}$ denote their symmetrized graph monomials.

Terminal stability conjecture. For each fixed $r\ge2$ there exists a finite list of square multigraphs $G_1,\ldots,G_N$, depending only on $r$ and not on $n$, such that for all $n\ge r$

$$\mathcal D_{n-r}^{(n)}=\sum_{\nu=1}^N c_\nu(n)\,\widetilde{G_\nu},\qquad c_\nu(n)\ge0,$$

where the coefficients $c_\nu(n)$ are rational functions, or equivalently polynomials after a common positive normalization, in $n$.

This is a conjectural strengthening of the square-graph cone problem, asserting uniform finite graph support for each fixed terminal order. The cases $r=2$ and $r=3$ are settled in the note; $r=4$ is identified as the next concrete target, and $r=5$ is reduced to an explicit quintic polynomial. The general assertion remains open.