Early polynomiality conjecture for the dimension of $W_{n,n+K}$

Fix $K\in\mathbb{N}$. Let $W_{n,n+K}$ be the representation considered in the paper, and let $p_K(x)$ and $N_K$ be as in the preceding corollary, so that $p_K(n)=\dim(W_{n,n+K})$ for every $n\geq N_K$. Polynomiality conjecture. In that corollary, one can take $N_K=2K$. This would make the polynomial dimension formula valid from the explicitly stated threshold $n\geq2K$, rather than only from some unspecified sufficiently large index. The source provides no resolution.