Principal minor sums of (AÂ +Â tB)m

The question is raised whether the sum of the k × k principal minors of the titled matrix is a polynomial (in t) with positive coefficients, when A and B are positive definite. This would generalize a conjecture made by Bessis-Moussa-Villani, as stated by E.H. Lieb and R. Seiringer [Equivalent forms of the Bessis-Moussa-Villani conjecture, J. Stat. Phys. 115 (2004) 185-190]. We give a variety of evidence for this further question, some of which is new.