Invariant element in the tensor product of rectangular Specht modules?

Question

Denote by $\mathfrak{S}_k$ the symmetric group on $k$ elements. Let $\lambda=(n^2\times n)=(n^2,\ldots,n^2)$ be a rectangular partition and $k=n^3$. Denote by $S_\lambda$ the Specht module corresponding to $\lambda$ and set $V:=S_\lambda^{\otimes 3}=S_\lambda\otimes S_\lambda\otimes S_\lambda$. There is a (unique, up to scalars) nonzero element $v\in V$ with $\sigma\cdot v=v$ for all $\sigma\in\mathfrak{S}_{n^3}$, this is something I can prove. However, I would like to know what that element looks like. Does anyone have any idea?