The biquadratic SOS rank conjecture for 3 by 3 forms

Let a biquadratic form be a homogeneous polynomial of degree two in each of two variable vectors, and let its SOS rank be the smallest number of squares of bilinear forms in a sum-of-squares representation. Let $\mathrm{BSR}(3,3)$ denote the maximum SOS rank among $3\times3$ biquadratic forms. The biquadratic SOS rank conjecture. For $3\times3$ biquadratic forms,

$$\mathrm{BSR}(3,3)=6.$$

The known bounds are $6\le \mathrm{BSR}(3,3)\le 8$, while the value $6$ is attained by a $3\times3$ simple biquadratic form. Determining whether every $3\times3$ biquadratic form has SOS rank at most six remains open.