Preliminaries and notation:
Let $n\in \mathbb{Z}_{>0}$ and $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_s)\vdash n$ be a partition. Given a Young diagram of shape $\lambda$, we can associate it with a Young tableau with $1,2,\dots,n$ filled in the Young diagram. A Young tableau is called standard if both the row and column entries in the Young diagram are strictly increasing.
Now let ${\rm Tab}(\lambda)$ be the set of all Young tableaux of shape $\lambda$, and let ${\rm Std}(\lambda)$ be the set of all standard Young tableaux of shape $\lambda$.
Given $\mathfrak{t}\in {\rm Tab}(\lambda)$, define the Young Symmetriser$$c_{\lambda}(\mathfrak{t}):=x_{\lambda}(\mathfrak{t})y_{\lambda}(\mathfrak{t}),$$where $x_{\lambda}(\mathfrak{t})=\sum_{\sigma\in R(\mathfrak{t})}\sigma, y_{\lambda}(\mathfrak{t})=\sum_{\sigma\in C(\mathfrak{t})}\epsilon(\sigma)\sigma$ with $R(\mathfrak{t})$ and $C(\mathfrak{t})$ being respectively the set of row stablisers and column stablises of $\mathfrak{t}$.
Let $h_\lambda={\prod_{i,j} h_{ij}^{\lambda}}$, where $h_{ij}^{\lambda}=\lambda_{i}+\lambda^{T}_{j}-i-j+1$ is the hook length of the $(i,j)$-box in the Young diagram of shape $\lambda$. $\lambda^{T}$ is the conjugate partition of $\lambda$.
Question
Does the following identity hold?$$ \sum_{\mathfrak t\in {\rm Tab}(\lambda)} c_{\lambda}(\mathfrak{t})=h_{\lambda} \sum_{\mathfrak t\in {\rm Std}(\lambda)} c_{\lambda}(\mathfrak{t}).$$
One can verify this directly for all partitions $\lambda\vdash n$ when $n=2,3$. But I have no idea for the general proof.
