I brought up a couple of combinatorial and number-theoretic items with this MO question. Now, I shall inquire on growth estimates. Recall $$K_r(n):=\prod_{\ell_1=1}^n\cdots\prod_{\ell_r=1}^n\left( 4\cos^2\left(\frac{\pi\ell_1}{2n+1}\right)+\cdots+4\cos^2\left(\frac{\pi\ell_r}{2n+1}\right)\right).$$

Question.Fix $r\geq3$. Can you provide an asymptotic (at least of first order) for $K_r(n)$, as $n\rightarrow\infty$?

**Example.** $K_1(n)=1$. It's also known (K.T.F.) that $K_2(n)\sim e^{\frac{4G}{\pi}n^2}$ where $G=$ Catalan's constant.