  In this note, we study minimal Lagrangian surfaces in $\mathbb{B}^4$ with
Legendrian capillary boundary on $\mathbb{S}^3$. On the one hand, we prove that
any minimal Lagrangian surface in $\mathbb{B}^4$ with Legendrian free boundary
on $\mathbb{S}^3$ must be an equatorial plane disk. One the other hand, we show
that any annulus type minimal Lagrangian surface in $\mathbb{B}^4$ with
Legendrian capillary boundary on $\mathbb{S}^3$ must be congruent to one of the
Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang
and Weng (Sci. China Math., 2020).
