The images of non-commutative polynomials evaluated on 2 × 2 matrices

Let p be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field K of any characteristic. It has been conjectured that for any n, the image of p evaluated on the set M_n(K) of n by n matrices is either zero, or the set of scalar matrices, or the set sl_n(K) of matrices of trace 0, or all of M_n(K). We prove the conjecture for n = 2, and show that although the analogous assertion fails for completely homogeneous polynomials, one can salvage the conjecture in this case by including the set of all non-nilpotent matrices of trace zero and also permitting dense subsets of M_n(K).