Functorial invariance of singular-category tangle presentations

Let $\cap_{i,n}$ and $\cup_{i,n}$ be the cap and cup functors between the relevant singular categories, and let $R_{i,n}$ be the cone of the adjointness morphism $\varepsilon_i\circ\mathcal R\Gamma_i\longrightarrow Id$. For a presentation $\alpha$ of a tangle as a composition of elementary tangles, let $f(\alpha)$ be the corresponding composition of these functors. Singular tangle-invariance conjecture. If $\alpha$ and $\beta$ are two presentations of the same tangle $t$, then $f(\alpha)$ and $f(\beta)$ should be isomorphic, up to shifts in the derived category. This is intended to produce functorial invariants of links and tangles; no proof is supplied in the stated passage.