Putnam 1986 B5

Let $f(x,y,z) = x^2+y^2+z^2+xyz$. Let $p(x,y,z), q(x,y,z)$, $r(x,y,z)$ be polynomials with real coefficients satisfying $$f(p(x,y,z), q(x,y,z), r(x,y,z)) = f(x,y,z).$$ Prove or disprove the assertion that the sequence $p,q,r$ consists of some permutation of $\pm x, \pm y, \pm z$, where the number of minus signs is 0 or 2.