The conformal-to-Einstein operator is a conformally invariant linear overdetermined differential operator whose non-vanishing solutions correspond to Einstein metrics within a conformal class. I will discuss the construction of a compatibility complex for this operator under natural genericity assumptions on the Weyl curvature in dimension $n\gt 4$, which implies at most one independent solution. The construction is based on a method I previously proposed in [arXiv:1805.03751], that generally applies to operators of finite type and leverages existing symmetries and geometric properties of the starting operator. (based on [arXiv:2602.08510])