Categorification conjecture for the positive half of a small quantum group

Let $cGamma$ be a Cartan datum of ADE type. Let $(R(cGamma),\partial)$ be the associated $p$-DG algebra, with the parameters of the differential $cpartial$ specified in the source's definition. Let $cmathbb{O}_p$ denote the coefficient ring for the integral form, and let the positive half of the small quantum group associated with $cGamma$ be its positive quantum-group subalgebra. Categorification conjecture. The $p$-DG algebra $(R(cGamma),\partial)$ categorifies an $cmathbb{O}_p$-integral form of the positive half of the small quantum group associated with $cGamma$. The authors note that the conjecture is vague because they do not specify which integral form over $cmathbb{O}_p$ is to be categorified; the source gives no resolution.