Sharp bounds for the valence of certain harmonic polynomials

In Khavinson and Swiatek (2002) it was proved that harmonic polynomials $z-\overline{p(z)}$, where $p$ is a holomorphic polynomial of degree $n > 1$, have at most $3n-2$ complex zeros. We show that this bound is sharp for all $n$ by proving a conjecture of Sarason and Crofoot about the existence of certain extremal polynomials $p$. We also count the number of equivalence classes of these polynomials.