Polynomial-growth conjecture for the dimension of $W_{n,n+K}$

For each $n,K\in\mathbb{N}$, let $W_{n,n+K}$ be the representation considered in the paper, and let $\dim(W_{n,n+K})$ denote its dimension. Dimension-growth conjecture. Fix $K\in\mathbb{N}$. There exist constants $c_K,C_K>0$ such that

$$c_K n^{2K}\leq\dim(W_{n,n+K})\leq C_K n^{2K}.$$

The preceding representation-theoretic bounds motivate this claim, which predicts that the dimension has exact polynomial order $n^{2K}$ for fixed $K$. The source provides no resolution.