Novak's positive-sem definiteness conjecture for trigonometric integration matrices

Let $n,d\ge 2$ and let $x_1,\dots,x_n\in\mathbb{R}^d$. Consider the $n\times n$ matrix whose $(j,k)$-entry is

$$\prod_{i=1}^d\frac{1+\cos(x_{j,i}-x_{k,i})}{2}-\frac{1}{n}.$$

Novak's conjecture. The matrix

$$\Bigl\{\prod_{i=1}^d\frac{1+\cos(x_{j,i}-x_{k,i})}{2}-\frac{1}{n}\Bigr\}_{j,k=1}^n$$

is positive semidefinite for all $n,d\ge 2$ and all choices of $x_1,\dots,x_n\in\mathbb{R}^d$. This is a conjecture on the intractability of numerical integration for trigonometric polynomials of degree at most one in each variable; the paper proves it as a consequence of its variant of Schur's product theorem. It also appeared as Open Problem 3 in Novak and Woźniakowski (2008).