Exact characterization conjecture for boundary hyperplane covers of quadratic sets

Let $B=\{\mathbf{x}:\|\mathbf{x}\|_2\leq 1\}$ and $C=\{\mathbf{x}:g(\mathbf{x})\leq 0\}$, where $g$ is a quadratic polynomial. Suppose that $g$ has one of the forms

$$g(\mathbf{x})=\alpha\left(\sum_{i=1}^n x_i^2-1\right)+h_1(\mathbf{x})h_2(\mathbf{x})$$

or

$$g(\mathbf{x})=\alpha\left(\sum_{i=1}^n x_i^2-1\right)+h_1(\mathbf{x}),$$

where $\alpha\neq 0$ and the $h_i$ are non-trivial affine functions. Exact characterization conjecture. Theorem~ holds as an if and only if: $B$ and $C$ have a Boundary Hyperplane Cover if and only if $g$ has one of these two forms, with the hyperplanes given by $h_i(\mathbf{x})=0$ as appropriate. The theorem supplies the sufficient direction; proving the converse requires algebraic tools to show that no other quadratic intersections have the Boundary Hyperplane Cover property.