Pure-derivative conjecture for derivations of polynomial subalgebras

Let $A\subseteq\mathbb{K}[x]$ be a subalgebra with spectrum containing $\alpha$, let $C=\{\alpha_1,\ldots,\alpha_s\}$ be its spectrum, and let $\alpha_j\sim\alpha$ mean that $\alpha_j$ belongs to the cluster containing $\alpha$. An $\alpha$-derivation is a derivation evaluated at $\alpha$ in the sense used for $A$. Pure-derivative conjecture. If $\alpha$ belongs to the spectrum, then every $\alpha$-derivation $D$ can be written as

$$D(f)=\sum_{i=1}^N\sum_{\alpha_j\sim\alpha}c_{ij}f^{(i)}(\alpha_j),$$

so only pure derivatives, of possibly different orders, at elements of the cluster containing $\alpha$ are needed. Equivalently, every finite-codimension subalgebra can be described using derivations of this form together with conditions $f(\alpha_i)=f(\alpha_j)$. The preceding results establish the analogous description using general linear combinations of derivative evaluations; the conjecture asserts that the mixed terms can always be replaced by pure derivatives.