On D. Haegele's approach to the Bessis-Moussa-Villani conjecture

The reformulation of the Bessis-Moussa-Villani conjecture given by Lieb and Seiringer asserts that the coefficient of t^r in the polynomial Trace[(A+tB)^p], with A and B positive semidefinite matrices, is nonnegative for all p and r. We propose a natural extension of a method of attack on this problem due to Haegele, and investigate for what values of p and r the method is successful, obtaining a complete determination when either p or r is odd.