As a consequence of the Littlewood-Richardson (LR) commuters coincidence and the Kumar-Torres branching model via Kushwaha-Raghavan-Viswanath flagged hives, we have solved the Lecouvey-Lenart conjecture on the bijections between the Kwon and Sundaram branching models for the pair \( (GL_{2n}(\mathbb C)) \),\( Sp_{2n}(\mathbb C) \) consisting of the general linear group \( GL_{2n}(\mathbb C) \) and the symplectic group \( Sp_{2n}(\mathbb C) \). In particular, thanks to the Henriques-Kamnitzer \( gl_n \)-crystal commuter, we have recognized that the left companion of an LR-Sundaram tableau is characterized by the Kwon symplectic condition. We now show that the construction of the left companion tableau of a LR-Sundaram tableau exhibits in fact the Kwon symplectic property.