Positive semidefiniteness conjecture for even-order symmetric quasi-double B0-tensors

Let $\mathcal{B}=(b_{i_1\cdots i_m})$ be a tensor of order $m$ and dimension $n$. For $i,j\in N$ with $i\neq j$, suppose that

$$(b_{i\cdots i}-\beta_i(\mathcal{B}))\left(b_{j\cdots j}-\beta_j(\mathcal{B})-\Delta_j^i(\mathcal{B})\right)\geq\left(\beta_j(\mathcal{B})-b_{ji\cdots i}\right)\Delta_i(\mathcal{B}).$$

Here, $\mathcal{B}$ is a quasi-double $B_0$-tensor when these inequalities hold for all such $i,j$. Positive semidefiniteness conjecture. Every even-order symmetric quasi-double $B_0$-tensor is positive semidefinite. The preceding results establish positive definiteness for even-order symmetric double and quasi-double $B$-tensors, but the method does not prove positive semidefiniteness under the weakened $B_0$-condition; this remains an open problem.