Geometric Borovik–Cherlin conjecture for differential fields

Let $\Delta=\{\delta_1,\dots,\delta_m\}$ be a set of $m$ commuting derivations in characteristic zero, and let $G$ be a strongly connected differential-algebraic group of $\Delta$-type $\ell$. Suppose that $G$ acts faithfully and transitively on a differential-algebraic variety $S$ of typical $\Delta$-dimension $d>0$. Let $\mathcal{F}$ be the constant field of some $m-\ell$ linearly independent derivations in $\operatorname{span}_{\mathcal{C}}(\Delta)$. Geometric Borovik–Cherlin conjecture. If $G$ acts Kolchin-generically $(d+2)$-transitively on $S$, then $(G,S)$ is isomorphic to the natural action of $\mathrm{PSL}_{d+1}(\mathcal{F})$ on $\mathbb{P}^d(\mathcal{F})$. This extends the finite-dimensional geometric formulation by replacing dimension with typical $\Delta$-dimension and allowing several commuting derivations; the source presents it as an expectation without a resolution.