On a conjecture about the μ-permanent

Let A = (a(ij)) be an n-by-n matrix. For any real mu, define the polynomial P-mu (A) Sigma(sigma is an element of Sn) a(1 sigma(1))...a(n sigma(n)) mu(l(sigma)), where l(sigma) is the number of inversions of the permutation sigma in the symmetric group S-n. We prove that P-mu(A) is a strictly increasing function of mu is an element of [- 1, 1], for a Hermitian positive definite nondiagonal matrix A, whose graph is a tree.