Complement determination conjecture for Brunnian theta-curves

Let $\Gamma_1$ and $\Gamma_2$ be Brunnian $\theta$-curves, meaning spatial $\theta$-curves in $S^3$ whose proper subgraphs are unknotted. Suppose their complements $S^3\setminus\Gamma_1$ and $S^3\setminus\Gamma_2$ are homeomorphic. Complement determination conjecture. There exists a homeomorphism of $S^3$ to itself taking $\Gamma_1$ to $\Gamma_2$. This predicts that a Brunnian $\theta$-curve is determined up to equivalence by its complement; its motivation is the uniqueness of Brunnian spines for genus $2$ handlebody knots, while the conjecture itself remains open.