Li–Wang–Weng conjecture on annular minimal Lagrangian surfaces

Let $\Sigma$ be an embedded annulus-type minimal Lagrangian surface in $\mathbb{B}^4$ with Legendrian capillary boundary on $\mathbb{S}^3$. Such a boundary has constant contact angle, where the contact angle $\theta$ is defined by $\nu=\sin\theta\,x+\cos\theta\,Jx$ along the boundary. Li–Wang–Weng's annulus conjecture. $\Sigma$ is one of the Lagrangian catenoids. This is a Lagrangian analogue of uniqueness conjectures for embedded free-boundary minimal annuli. The paper gives an affirmative answer and proves the statement without the embeddedness assumption, so the conjecture is solved.