A-polynomial identities in Grassmann and in matrix algebras

The A-identities were first studied (although implicitly) around 1955 by  Kostant. Their more systematic study was started some 10 years ago by Regev. Later on Henke and Regev studied these identities in the Grassmann algebra.
An A-monomial of degree n is an even permutation of the noncommutative variables x_1 to x_n; an A-polynomial of degree n is a linear combination of such monomials in the free associative algebra.
Henke and Regev proposed two conjectures concerning the A-identities satisfied by the Grassmann algebra, and the minimal degree of an A-identity for the matrix algebras. I shall discuss these two conjectures. The first turns out to be true while the second fails.