Consider $T$ to be a standard Young tableau of shape $\lambda$ (in English notation). The descent set of $T$, $Des(T)$, is defined to the set of all positive integers $i$ such that $i+1$ lies strictly south (and weakly west) of $i$ in $T$.

A flip on $T$ is defined to be a move where we exchange the positions of $i$ and $i+1$ in $T$ if $i\in Des(T)$, provided that $i$ and $i+1$ are not the same column. Let $T'$ be another standard Young tableau obtained by doing a sequence of flips starting from $T$.

$\textbf{Question}$: Is it true that $Des(T)\neq Des(T')$?

It seems to be true from the examples I worked out, and seemed to be sort of result that would be present in literature. If this is indeed the case, I'd appreciate any reference that states this.

Thanks.