The algebraic reformulation of the BMV conjecture is equivalent to a family of dimensionfree tracial inequalities involving positive semidefinite matrices. Sufficient conditions for these to hold in the form of algebraic identities involving polynomials in noncommuting variables have been given by Markus Schweighofer and the second author. Later the existence of these certificates has been settled for all but one case, which is resolved in this short note.
K. Cafuta, I. Klep, and J. Povh. A note on the nonexistence of sum of squares certicates for the Bessis-Moussa-Villani conjecture. J. math. phys., 51(8):083521, 10, 2010.
View On the nonexistence of sum of squares certificates for the BMV conjecture