The pillowcase perturbation conjecture for 2-tangle character varieties

Let $(Y,T)$ be a 2-tangle in the 3-ball or in a homology 3-ball. Write $R_\pi(Y,T)$ for the perturbed traceless character variety and let

$$L_1:R_\pi(Y,T)\to R(S^2,\{a,b,c,d\})$$

be its restriction map. A pillowcase perturbation conjecture asserts that there exist arbitrarily small holonomy perturbations $\pi$ such that $R_\pi(Y,T)$ is a compact $1$-manifold with two boundary points and $L_1$ is a restricted immersed $1$-manifold on each component in the sense of the paper's definition. This would make the Lagrangian-Floer construction applicable to arbitrary 2-tangles, whereas the claim is presently supported by results for particular families and examples but remains unproved in general.