Invariants and dualities of a certain parabolic group

In this note, I will prove a conjecture in \cite{BYY}, which is related to the invariants of a maximal parabolic subgroup of ${\text{GL}}_{n+1}$. Consequently, the natural tensor invariants of this typical maximal parabolic subgroup of ${\text{GL}}_{n+1}$ are determined when $n\ge r$.