Upper bounds of dual flagged Weyl characters

For a subset D of boxes in an $n\times n$ square grid, let ${\chi }_D\left(x\right)$ denote the dual character of the flagged Weyl module associated to D. It is known that ${\chi }_D\left(x\right)$ 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 ${\chi }_D\left(x\right)$. Mészáros, St. Dizier and Tanjaya conjectured that ${\chi }_D\left(x\right)$ attains the upper bound if and only if D avoids a certain single subdiagram. We provide a proof of this conjecture.