Trace functions as Laplace transforms

We study trace functions on the form t→Trf(A+tB) where f is a real function defined on the positive half-line, and A and B are matrices such that A is positive definite and B is positive semidefinite. If f is non-negative and operator monotone decreasing, then such a trace function can be written as the Laplace transform of a positive measure. The question is related to the Bessis-Moussa-Villani conjecture.