Wang–Sun formula for quasipolarities when the modulus is even

Let $k$ be even, let $\zeta$ be a primitive $2k$-th root of unity, and let $Q_k$ denote the set of quasipolarities. For each permutation $\pi\in Q_k$, consider the product over $j=0,\ldots,n-1$ appearing below. Wang–Sun conjecture for even $k$. If $\zeta$ is a $2k$-th primitive root of unity, then

$$\sum_{\pi\in Q_{k}}\prod_{j=0}^{n-1}\frac{1+\zeta^{j-\pi(j)}}{1-\zeta^{j-\pi(j)}} = \sum_{\pi\in Q_{k}}\prod_{j=0}^{n-1}[j-\pi(j)\neq k].$$

This conjecture proposes an evaluation of the Wang–Sun sum in the even-$k$ case, extending the preceding vanishing theorem for odd $k$. The supplied text gives no resolution or further evidence beyond describing the formula as reasonable, so its status remains open.