Reconstruction of noncommutative Grothendieck–Witt motives without replacing objects

Let $NcS$ be the category whose objects are dg-categories without additional structures, and let $\widetilde{NcS}(S)$ and $DM^{\mathcal{GW}}_{nc}(k)$ be the categories constructed using dg-categories with duality. Reconstruction question. Can one reconstruct the category $\widetilde{NcS}(S)$ or $DM^{\mathcal{GW}}_{nc}(k)$ without replacing the objects of $NcS$, and by replacing morphisms only? This asks whether the duality data used in the construction can be encoded entirely in morphisms rather than by changing the objects; the source poses it as a question and supplies no answer.