The sharpness conjecture for triviality of cochains on spheres

Let $n\geq 1$ and let $A$ be an $\mathbb{E}_{n+2}$-algebra in $\operatorname{Sp}$. For a pointed space $X$, write $C^*(X;A)$ for the $A$-valued cochains on $X$, regarded as an augmented $\mathbb{E}_{n+1}$-algebra over $A$; an augmented algebra is $\mathbb{E}_{n+1}$-trivial over $A$ when it is a trivial square-zero extension in the corresponding category of augmented algebras. The sharpness conjecture. $C^*(S^n;A)$ is $\mathbb{E}_{n+1}$-trivial over $A$ if and only if $A$ is a $\mathbb{Q}$-algebra. The preceding results show that the cochains on spheres are always $\mathbb{E}_n$-formal, while the rational case gives $\mathbb{E}_{n+1}$-triviality. The conjecture asserts that rational ring spectra are the only cases in which this stronger conclusion holds.