The SOS conjecture for symmetric biquadratic M-tensors

Let $\mathcal{A}$ be a symmetric biquadratic M-tensor: a symmetric biquadratic tensor that can be written as $\mathcal{A}=\alpha\mathcal{I}-\mathcal{B}$, where $\mathcal{I}$ is the M-identity tensor, $\mathcal{B}$ is nonnegative, and $\alpha\geq\lambda_{\max}(\mathcal{B})$. Here an SOS tensor is one whose associated biquadratic form is a finite sum of squares of bilinear forms. SOS conjecture for symmetric biquadratic M-tensors. Every symmetric biquadratic M-tensor is an SOS tensor. The source notes that such tensors are PSD, but says that proving the SOS assertion would require new techniques connecting M-eigenvalues with matrix eigenvalues.