The Bottman–Wehrheim immersed composition conjecture

Let $(\underline{L},{\bf b})$ be a generalized immersed Lagrangian correspondence from $M_0$ to $M_k=M_0$, equipped with bounding cochains. Suppose that for some $i$, $L_{(i-1)i}$ and $L_{i(i+1)}$ can be composed, and let $(\underline{L}',{\bf b}')$ be obtained by replacing these two consecutive correspondences with

$$\bigl(L_{(i-1)i}\circ L_{i(i+1)},8(b_{(i-1)i},b_{i(i+1)})\bigr).$$

Bottman–Wehrheim conjecture. There is an isomorphism

$$\mathit{HF}(\underline{L},{\bf b})\cong\mathit{HF}(\underline{L}',{\bf b}').$$

This is the proposed extension of the embedded quilted Floer composition theorem to immersed correspondences; figure-eight bubbling and the associated bounding cochain are precisely the analytic obstacles, so the assertion remains open in the context described.