Reformulating the SOS conjecture via Newton-Okounkov bodies shows the minimal rank occurs at extreme points of the body, and for diagonal polynomials these reduce to finitely many rational points.
Ebenfelt, On the HJY Gap Conjecture in CR Geometry vs
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A Newton-Okounkov Body Viewpoint on the SOS Conjecture
Reformulating the SOS conjecture via Newton-Okounkov bodies shows the minimal rank occurs at extreme points of the body, and for diagonal polynomials these reduce to finitely many rational points.