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In this note, we study minimal Lagrangian surfaces in B 4 with Legendrian capillary boundary on S 3. On the one hand, we prove that any minimal Lagrangian surface in B 4 with Legendrian free boundary on S 3 must be an equatorial plane disk. On the other hand, we show that any annulus type minimal Lagrangian surface in B 4 with Legendrian capillary boundary on S 3 must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng [12]. [ABSTRACT FROM AUTHOR]