For a subset $D$ of boxes in an $n\times n$ square grid, let $χ_{D}(x)$ denote the dual character of the flagged Weyl module associated to $D$. It is known that $χ_{D}(x)$ specifies to a Schubert polynomial (resp., a key polynomial) in the case when $D$ is the Rothe diagram of a permutation (resp., the skyline diagram of a composition). One can naturally define a lower and an upper bound of $χ_{D}(x)$. M{é}sz{á}ros, St. Dizier and Tanjaya conjectured that $χ_{D}(x)$ attains the upper bound if and only if $D$ avoids a certain subdiagram. We provide a proof of this conjecture.