Conjecture on the necessary ambient-dimension dependence of covering numbers

Let $V$ be a $(K,n)$ regular set in $\mathcal{B}^N$, and let $\mathcal{N}(V,\eps)$ denote its covering number at scale $\eps$. Covering-number dependence conjecture. There exists a $(K,n)$ regular set $V$ in $\mathcal{B}^N$ with

$$\mathcal{N}(V,\eps)=\Omega(N^n).$$

This would show that an $N^1$ dependence is necessary in the relevant covering-number bound, improving the currently established $N^{3/2}$ dependence. The source does not provide evidence resolving the conjecture.