Geometric sign conjecture for the phase-two extrapolation parameters

Assume $f obreak\text{ belongs to } C^{1,1}_\nu(\mathbb{R}^n)$. Let $\hat{f}$ be the linear function that interpolates $f$ at any set of $n+1$ affinely independent vectors $\Theta=\{\mathbf{x}_1,\dots,\mathbf{x}_{n+1}\}\subset\mathbb{R}^n$. Let $\mathbf{x}$ be any vector in $\mathbb{R}^n$, and let $\{\mu_{ij}\}_{i\in\mathcal{I}_+,\ j\in\mathcal{I}_-}$ be the parameters defined in Theorem 2.3 of the source paper. Geometric sign conjecture. If the simplex $\operatorname{conv}(\Theta)$ has no obtuse angle at any vertex, meaning

$$(\mathbf{x}_j-\mathbf{x}_i)\cdot(\mathbf{x}_k-\mathbf{x}_i)\geq 0\quad\text{for all }i,j,k=1,2,\dots,n+1,$$

then all parameters $\{\mu_{ij}\}$ are non-negative for every $\mathbf{x}\in\mathbb{R}^n$. If the simplex has at least one obtuse angle at its vertices, then there is a non-empty subset of $\mathbb{R}^n$ such that, whenever $\mathbf{x}$ belongs to that subset, at least one element of $\{\mu_{ij}\}$ is negative. A general formula for the sharp bound on the function approximation error of linear interpolation and extrapolation remains open; the conjecture is motivated by numerical experiments and complements the proved sufficient condition that all $\mu_{ij}$ are non-negative.