The universal framed prodeformation conjecture

Let $A$ be a connective dga and let $S$ be a one-dimensional $A$-module. Let $E\coloneqq {\mathrm{\normalfont\mathbb{R}}}\mathrm{End}_A(S)$ be the derived endomorphism dga of $S$. Write $E^!$ for the corresponding deformation base, and let \text{\raisebox{.15ex}{\mathscr{D}}}\mathrm{ef}^{\mathrm{fr}}_A(S)(E^!) denote the set of framed deformations of $S$ over $E^!$. The map from this set to \widehat{\text{\raisebox{.15ex}{\mathscr{D}}}\mathrm{ef}}^{\text{\raisebox{-1.1ex}{\mathrm{fr}}}}_A(S)(B^\sharp E) sends a deformation over $E^!$ to its associated framed prodeformation over $B^\sharp E$. Universal framed prodeformation conjecture. The image of the deformation E^!\in \text{\raisebox{.15ex}{\mathscr{D}}}\mathrm{ef}^{\mathrm{fr}}_A(S)(E^!) under this map is the universal framed prodeformation of $S$ as an $A$-module. This identifies the explicit deformation $E^!$ with the universal framed prodeformation in the one-dimensional case.