Equivalence with the motive category built from paired dg-functors

Let $NcS^{\cdot-\cdot}$ be the category whose objects are dg-categories and whose morphisms are pairs of dg-functors $F,G$ equipped with a natural equivalence $F\simeq G$. Let $\mathcal P(NcS^{\cdot-\cdot})$ denote the category of presheaves on it, and let $L_{nc-nis}$ and $L_{\mathbb A^1,nc}$ be the indicated noncommutative Nisnevich and affine-line localisations. Equivalence conjecture. The $(\infty,1)$-category $DM^{\mathcal{GW}}_{nc}(k)$ is equivalent to

$$L_{nc-nis}L_{\mathbb A^1,nc}\mathcal P(NcS^{\cdot-\cdot}).$$

This is presented as a proposed answer to the preceding reconstruction question, based on encoding the information of pullback and adjoint functors through paired dg-functors and a natural equivalence; no proof or resolution is supplied.