Symplectic linear independence conjecture

Let $\mathcal{C}_m(\mathbb{R}^3)$ consist of tuples $\mathbf{x}=(\mathbf{x}_1,\ldots,\mathbf{x}_m)$ such that $\mathbf{x}_a\neq\mathbf{0}$ and $\mathbf{x}_a\pm\mathbf{x}_b\neq\mathbf{0}$ for $a<b$. Set $I=\{1,\ldots,m\}\cup\{\bar{1},\ldots,\bar{m}\}$ with order $1<\bar{1}<\cdots<m<\bar{m}$, define $\mathbf{x}_a$ as given and $\mathbf{x}_{\bar a}=-\mathbf{x}_a$, choose Hopf lifts of all differences $\mathbf{x}_\beta-\mathbf{x}_\alpha$, and set $p_{\alpha\beta}(t)=u_{\alpha\beta}t-v_{\alpha\beta}$ and

$$p_\alpha(t)=\prod_{\beta\in I\setminus\{\alpha\}}p_{\alpha\beta}(t).$$

Symplectic LIC. The polynomials $p_\alpha$, for $\alpha\in I$, are linearly independent over $\mathbb{C}$. This is the symplectic analogue of the Atiyah-Sutcliffe LIC for the root system associated with $Sp(m)$; the source gives no resolution.