Conjecture on definiteness of separable symmetric tensors

Let $\textbf{v}_{1},\textbf{v}_{2}, \ldots,\textbf{v}_{m}\in \mathbb R^{n}$ with $n\ge m$ be linearly independent vectors, and let

$$\mathcal{A}=\textbf{v}_{1}\vee \textbf{v}_{2}\vee \ldots \vee\textbf{v}_{m}.$$

Definiteness conjecture. Then $\mathcal{A}$ is neither a positive semidefinite tensor nor a negative semidefinite tensor.

This conjectures that the conclusion established for separable symmetric tensors of order two extends to all orders $m\ge 2$. The source gives no resolution beyond the order-two lemma, so the general statement remains open.