Bahri–Xu linear algebraic conjecture

Let $x_1,\dots,x_p$ with $p\geq 2$ be distinct vectors in $\mathbb R^m$. Define the symmetric matrix $A=(a_{ij})_{1\leq i,j\leq p}$ by

$$a_{ij}=\begin{cases}0,&i=j,\\\\ \dfrac{1}{|x_i-x_j|},&i\ne j. \end{cases}$$

For $k=1,\dots,p$, let $\dfrac{\partial A}{\partial x_k}=\left(\dfrac{\partial a_{ij}}{\partial x_k}\right)_{1\leq i,j\leq p}$, regarded as a vector-valued $p\times p$ matrix, and let $U=(u_1,\dots,u_p)^{\mathrm T}\in\mathbb R^p$. Bahri–Xu's linear algebraic conjecture. There is a positive constant $c=c(p,m)$ depending only on $p$ and $m$ such that

$$|AU|^2+\sup_{1\leq i\leq p}\left|U^{\mathrm T}\left(\frac{\partial A}{\partial x_i}\right)U\right|\geq c(p,m)\sum_{\substack{i,j\\\j\ne i}}\frac{u_j^2}{|x_i-x_j|^2}.$$

This is the purely linear-algebraic assumption introduced by Bahri and Xu in connection with asymptotic expansions for the Yamabe functional and the Morse lemma at infinity. The supplied source does not establish the assertion or provide evidence of its resolution, so its status remains open.