The symplectic left companion of a Littlewood-Richardson-Sundaram tableau and the Kwon property

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 use the construction of the left Gelfand-Tsetlin pattern, or left companion tableau, of an LR-Sundaram tableau to exhibit the Kwon symplectic property, a mirror of the flag on its right companion tableau. This is equivalent to the restriction of the Berenstein-Gelfand-Zelevinsky LR model, on the interpretation of Gelfand-Tsetlin patterns, to symplectic Gelfand-Tsetlin patterns as left companions of LR-Sundaram tableaux.