Uniqueness conjecture for derivations on the Adams $E_2$ page of the sphere

Let $E_2$ denote the Adams $E_2$ page for the sphere, viewed with its algebraic structure, and let its differentials be regarded as derivations. Uniqueness conjecture. The space of derivations on $E_2$ is $1$-dimensional. Equivalently, there is only one nontrivial set of differentials on $E_2$ compatible with its algebraic structure. This would imply that the algebraic structure of the Adams $E_2$ page determines the differentials uniquely, resolving the ambiguities that remain in the authors' automated differential computations. The source provides no resolution of the conjecture.