The chain-level $\mathfrak{gl}_N$ link invariant functoriality conjecture

Let $\mathbf{Links}_\infty(\mathbb{R}^3)$ be the $(\infty,1)$-category of links in $\mathbb{R}^3$, link cobordisms in $\mathbb{R}^3 \times I$, isotopies, and higher isotopies. Let $\mathrm{Ch}^b(\mathbb{Z}\text{-}\operatorname{gpmod})$ be the $(\infty,1)$-category of bounded chain complexes, chain maps, homotopies, and higher homotopies. Write $\operatorname{CKhR}_N$ for the chain-level $\mathfrak{gl}_N$ link invariant from the cited construction.

Chain-level $\mathfrak{gl}_N$ functoriality conjecture. The chain-level $\mathfrak{gl}_N$ link invariant arises as the truncation of an $\mathbb{E}_3$-monoidal functor

$$\operatorname{CKhR}_N \colon \mathbf{Links}_\infty(\mathbb{R}^3) \longrightarrow \mathrm{Ch}^b(\mathbb{Z}\text{-}\operatorname{gpmod}).$$

This would promote the usual chain-level link invariant to homotopy-coherent functoriality, including coherence for isotopies and higher isotopies. The parser supplies no evidence that the statement has been proved or disproved, so its status remains open.