The Novikov kernel conjecture for universal geometric operations

Consider the universal map $\Phi_{{\text{\rm\tiny geo}}}^{\mathrm{Novikov}}$ induced by $\Phi_{{\text{\rm\tiny geo}}}$ after taking the operad quotient

$$\operatorname{PreLie}\twoheadrightarrow\mathrm{Novikov}$$

in the construction given by this approach. Here $\hat{\Phi}_{{\text{\rm\tiny geo}}}^{\mathrm{Novikov}}$ denotes the corresponding map on the relevant completed spaces, and iterations of covariant derivatives are the operations obtained by repeatedly applying covariant differentiation. Kernel conjecture. The kernel $\ker\hat{\Phi}_{{\text{\rm\tiny geo}}}^{\mathrm{Novikov}}$ is the linear span of iterations of covariant derivatives. This conjecture proposes that, in dimension one, no additional exotic operations occur beyond those generated by iterated covariant derivatives; the surrounding discussion notes that no exotic operations are known in that dimension, while the analogous question in lower dimensions is otherwise not fully characterized.