Tameness conjecture for the Lagrangians in the twisted character variety

Let $\Sigma^{\#}$ be the punctured Heegaard surface and let $X_{\operatorname{tw}}(\Sigma^{\#})$ be its twisted character variety. Let $L^{\#}_0,L^{\#}_1\subset X_{\operatorname{tw}}(\Sigma^{\#})$ be the Lagrangians associated with the two handlebodies. Tameness conjecture. The Lagrangians $L^{\#}_0$ and $L^{\#}_1$ are tame in the sense of Sikorav. Establishing this condition would help control holomorphic strips escaping to infinity and enable the construction of the proposed non-compact Lagrangian Floer homology. The source gives no evidence that the claim has been proved or refuted.