Let $\lambda$ be an integer partition, denote the number of Standard Young Tableaux of shape $\lambda$ by $f_{\lambda}$. This number is computed by the formula
$$f_{\lambda}=\frac{n!}{\prod_{u\in\lambda}h_u}$$
where $h_u$ is a hook length. It is also well-known that
$$\sum_{\lambda\vdash n}f_{\lambda}^2=n!\tag1$$
Recall the notation for the **content** of a cell $u=(i,j)$ in a partition is $c_u=j-i$.

Question.Is this true? $$\sum_{\lambda\vdash n}f_{\lambda}^2\prod_{u\in\lambda}(t+c_u)=n!\,t^n.$$

**NOTE.** An affirmative answer would imply (1): divide both sides by $t^n$ and take the limit $t\rightarrow\infty$.