On the $A_{\mathrm{\infty }}$-extension of Bar-Natan multicurves

Kotelskiy, Watson and Zibrowius show that the Bar-Natan invariant $\text{unicode}x0414(T)$ associated to a $4$-ended oriented tangle $T$ extends to a twisted complex $\text{unicode}{x0414}^{\mathrm{\infty }}(T)$ over an $A_{\mathrm{\infty }}$-enhancement of the Bar-Natan algebra. We prove that $\text{unicode}{x0414}^{\mathrm{\infty }}(T)$ itself is a tangle invariant, settling a conjecture from their work. We deduce this from an invariance statement for a \emph{matrix multifactorization} $\mathcal{M}(T)$, generalizing a construction of Ballinger to the coefficient ring $\mathbb{Z}[G]$. This multifactorization is well defined up to a notion of 1-homotopy equivalence. The proof rests on explicit formulas for special deformation retracts of Koszul matrix factorizations, which realize delooping in this setting and may be of independent interest.