Khovanov invariants via Fukaya categories: the tangle invariants agree

Given a pointed 4-ended tangle $T\subset D^3$, there are two Khovanov theoretic tangle invariants, $\text{unicode}{1044}_1(T)$ from [arXiv:1910.1458] and $L_T$ from [arXiv:1808.06957], which are twisted complexes over the Fukaya category of the boundary 4-punctured sphere $(S^2,4$\text{pt}$)=\partial (D^3, T)$. We prove that these two invariants are the same.