Symplectic invariant-tensor character conjecture

Let $n\ge 1$, let $\mathfrak{g}=\mathfrak{sp}(2n)$, and let $V$ be the vector representation. For a partition $\mu$, write $\overline{2\mu}$ for the partition obtained by doubling each part, and let $s_\lambda$ denote the Schur function indexed by $\lambda$. Let $N(\otimes^{2r}V)$ be the invariant tensors with its natural $S(2r)$-representation structure. Symplectic invariant-tensor character conjecture. For $r\ge 1$, the character of the space of invariant tensors of $\otimes ^{2r}V$ as a representation of $S(2r)$ is

$$\sum_{\mu} s_{\overline{2\mu}}$$

where the sum is over partitions $\mu$ such that $\mu$ has at most $2n$ parts and $|\mu|=r$. This predicts the symmetric-group character of symplectic invariant tensors; the supplied text does not state whether the formula is resolved.