The naturality conjecture for the earring endofunctor

Let $P^*$ be the pillowcase, let $\mathcal{W}^\star(P^*)\simeq\operatorname{Tw}\mathcal{B}$ be the indicated subcategory, and let $\mathcal{N}$ denote the curve-to-twisted-complex correspondence. Let $\mathcal{X}(\operatorname{NAT}_s)$ be the hypothetical $A_\infty$ endofunctor induced by the immersed Lagrangian correspondence $\operatorname{NAT}_s$. Naturality conjecture. For every curve $L$ not touching the top-left corner of the pillowcase, the displayed diagram commutes; equivalently,

$$\mathcal{X}(\operatorname{NAT}_s)\colon\mathcal{W}(P^*)\longrightarrow\mathcal{W}(P^*)$$

restricted to $\mathcal{W}^\star(P^*)\simeq\operatorname{Tw}\mathcal{B}$ equals the endofunctor $\mathcal{H}=[\mathbf{I}\to\mathbf{I}]$. The computations for the trivial tangle and a three-twist tangle provide evidence, while the general statement remains conjectural.