Nonexistence of a filling inducing the nontrivial augmentation of the standard Legendrian unknot

Let $U$ be the standard Legendrian unknot, whose unique Reeb chord is denoted by $b$. A conical Legendrian filling $\Sigma$ of $U$ is equipped with a $1$-graded augmentation $\alpha:\mathcal{A}(\Sigma)\rightarrow\mathbb{Z}/2$, inducing an augmentation $\epsilon_{(\Sigma,\alpha)}$ of $\mathcal{A}(U)$. Filling nonexistence conjecture. There is no such conical Legendrian filling $\Sigma$ and augmentation $\alpha$ for which $\epsilon_{(\Sigma,\alpha)}(b)=1$. The preceding theorem shows that every augmentation can be realized using a conical Legendrian cobordism from the standard unknot, but the case in which the augmentation on the unknot sends $b$ to $1$ is not known to arise from a filling; the conjecture asserts that it never does.