Quadratic-bound reconstruction conjecture for tableau minors

Let $k\geq 2$ and let $T\in\operatorname{YT}(n)$, where $n\geq k^2+2k$. Here $\operatorname{M}_k(T)$ denotes the collection of $k$-minors of the standard Young tableau $T$. Reconstruction conjecture. The collection $\operatorname{M}_k(T)$ determines $T$. The quadratic bound is suggested by the cases checked by computer, and the conjecture concerns reconstructibility from $\operatorname{M}_k(T)$ for general $k$.