The five-square conjecture for 3 × 3 symmetric PSD biquadratic forms

Let $P(\mathbf{x},\mathbf{y})$ be a $3 \times 3$ symmetric PSD biquadratic form, so $P(\mathbf{x},\mathbf{y})\geq 0$ for all $\mathbf{x},\mathbf{y}\in\mathbb{R}^3$. A bilinear form is a polynomial linear in each of $\mathbf{x}$ and $\mathbf{y}$ separately. Five-square conjecture. Every $3 \times 3$ symmetric PSD biquadratic form can always be expressed as the sum of at most five squares of bilinear forms. The paper proves the weaker SOS statement for all symmetric PSD biquadratic forms and notes that the bound five is suggested by explicit decompositions and the known rank of an interior example, but it remains unproved.