A result on the Gelfand-Kirillov dimension of representations of classical groups

Let $(G,G')$ be the reductive dual pair $(O(p,q),Sp(2n,\mathbb{R}))$. We show that if $\pi$ is a representation of $Sp(2n,\mathbb{R})$ (respectively $O(p,q)$) obtained from duality correspondence with some representation of $O(p,q)$ (respectively $Sp(2n,\mathbb{R})$), then its Gelfand-Kirillov dimension is less than or equal to
$(p+q)(2n-\frac{p+q-1}{2})$ (respectively $2n(p+q-\frac{2n+1}{2})$).