Classification conjecture for Lagrangian disks with conformal Maslov form

Let $D$ be a disk immersed in $4\mathbb{R}^4$. Assume that the immersion is Lagrangian, has conformal Maslov form, and has Legendrian capillary boundary on $4\mathbb{S}^3$. Classification conjecture. Any immersed Lagrangian disk in $4\mathbb{R}^4$ with conformal Maslov form and Legendrian capillary boundary on $4\mathbb{S}^3$ must be either totally geodesic or one of the Whitney spheres given in Example 2.9. The conjecture is a boundary analogue of the classification of Lagrangian spheres with conformal Maslov form as Whitney spheres; it is proved when the boundary is assumed to be a circle on $4\mathbb{S}^3$.