Pfister's theorem fails in the Hermitian case

We show that the Hermitian analogue of a famous result of Pfister fails. To do so we provide a Hermitian symmetric polynomial $r$ of total degree 2d such that any non-zero multiple of it cannot be written as a Hermitian sum of squares with fewer than $d+1$ squares.
Comment: 7 pages, to appear in Proc. Amer. Math. Soc., improved exposition in the proof