The Bottman–Wehrheim correspondence conjecture for the earring tangle

Let $P^*$ be the pillowcase, let $u_s$ and $v_\delta$ be the immersed Lagrangian correspondences associated to the earring tangle and its model, respectively, and let $A$ be a $\delta$-arc-type curve. Bottman–Wehrheim correspondence conjecture. For sufficiently small $\delta$, the immersions $u_s$ and $v_\delta$ are Hamiltonian regularly homotopic in $(P^*)^-\times P^*$ for all sufficiently small $s>0$. Moreover, $u_s$ is transverse to every $\delta$-arc-type curve and sends each such curve to a figure eight curve. This would strengthen the preceding proposition from homology figure eight curves to figure eight curves and supply the geometric input needed for the earring correspondence.