The interval conjecture for shapes in K-Knuth classes

Let $T_1,T_2,\dots,T_k$ be a $K$-Knuth class of straight tableaux, and let $T_i$ have shape $\lambda_i$. Write

$$\Sigma=\{\lambda_1,\lambda_2,\dots,\lambda_k\}.$$

Order Young diagrams by inclusion, and let $[\lambda_i,\lambda_j]$ denote the interval in Young's lattice. Shape-interval conjecture. If $\lambda_i,\lambda_j\in\Sigma$, then

$$[\lambda_i,\lambda_j]\subseteq\Sigma.$$

This asserts that all shapes between any two shapes occurring in a class also occur. It was verified in the supplied text for $K$-Knuth classes on $[n]$ with $n\leq 7$, but remains open in general.