Dimension conjecture for ordinary-derivative derivations

Let $A\subseteq\mathbb{K}[x]$ be a finite-codimension subalgebra, let $k_\alpha$ be the dimension of the vector space of all $\alpha$-derivations of $A$, and let $\mathcal{D}^A_\alpha$ be the subspace of those $\alpha$-derivations expressible as linear combinations of ordinary derivative evaluations $f'(\beta),f”(\beta),\ldots$ for points $\beta$ in the cluster of $\alpha$. Dimension conjecture. For every $\alpha$,

$$\dim\mathcal{D}^A_\alpha=k_\alpha.$$

This equality would show that all $\alpha$-derivations are generated by ordinary derivatives at points in the relevant cluster. The authors present it as a strengthening useful for proving the main pure-derivative conjecture; it is proved in the paper only for some special classes of subalgebras.