 Solution

The equality $h\{t\}=h^{\ast }\{t^{\ast }\}$ says that $h(i)$ and $h^{\ast }(i)$ lie in corresponding rows for every entry $i$. For $i$ in the leftmost column, both $h(i)$ and $h^{\ast }(i)$ remain in that column. Corresponding rows therefore force $h(i)=h^{\ast }(i)$.

Remove the leftmost entry of every row and repeat the argument on the shortened tableaux. Induction across the columns gives equality on every entry, so $h=h^{\ast }$.