A surprising identity: $\det[\cos\pi\frac{jk}n]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}$

On the basis of my computation, here I pose my following conjecture involving the cosine function.

Conjecture. For any positive integer $$n$$, we have the identity $$\frac1{2n}\det\left[\cos\pi\frac{jk}n\right]_{0\le j,k\le n}=\det\left[\cos\pi\frac{jk}n\right]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}.$$

This is a part of Conjecture 5.7 in my preprint arXiv:1901.04837. The paper contains more similar conjectures.

Any ideas towards a solution of the conjecture?