A Problem In Linear Algebra

I have the following problem, I do not know the answer.

Assume that d and n are natural numbers and define an n times n matrix A by its entries

a_{ij} = ( \prod_{l=1}^d \cos^2 (x_i^l - x_j^l) ) - 1/n .

The real numbers x_i^l are arbitrary.

Prove or disprove the following CONJECTURE:

The matrix A is positive semidefinite, i.e., the eigenvalues are nonnegative.


I know that the conjecture is true for $n \ge 2^d$. So the interesting case would be $n < 2^d$.

The problem came up when I wanted to prove some error bounds for cubature formulas. I made many numerical tests and, of course, did not find a counterexample of the conjecture. I want to write a paper concerning the consequences of my conjecture, this paper will be available in May 1998.

This paper now is finished:
Intractability Results for Positive Quadrature Formulas
and Extremal Problems for Trigonometric Polynomials.
J. Complexity 15 (1999), 299-316.

Please let me know if you have comments, questions, or even a solution.

More than ten years later: The problem is still not solved but there is a very interesting paper by Aicke Hinrichs and Jan Vybiral with related (and more general) problems and conjectures: "On positive positive-definite functions and Bochner's theorem". Journal of Complexity 27 (2011), 264-272.