Every augmentation lifts to a curved augmentation

Let $(\mathcal{A},\partial)$ be the Chekanov–Eliashberg algebra of a Legendrian, let $(\mathcal{L},d)$ be its LSFT algebra, and let $\epsilon:\mathcal{A}\to\Bbbk$ be an augmentation. A $c$-curved augmentation is a map $\tilde{\epsilon}:\mathcal{L}\to\Bbbk\llbracket\eta\rrbracket$ lifting $\epsilon$ and having curvature parameter $c\in\Bbbk$. Every-augmentation lifting conjecture. Every augmentation lifts to a $c$-curved augmentation, for any $c\in\Bbbk$. Computational examples suggest that such lifts exist generally, extending the canonical lift in the case $c=0$; whether this holds for all augmentations remains open.