Bremner–Madariaga conjecture on the quadraticity of the second Veronese power of PreLie

Let $\operatorname{PreLie}$ be the operad of pre-Lie algebras, and let $\operatorname{PreLie}^{[2]}$ denote its second Veronese power. The preceding computations show that all relations of weight at most two between the natural ternary generators follow from the displayed quadratic relations, and no cubic relations beyond the quadratic consequences are known. Bremner–Madariaga conjecture. The operad $\operatorname{PreLie}^{[2]}$ is quadratic. This conjecture is formulated after computations showing that there are no cubic relations that do not follow from the quadratic ones; its resolution is not given here.