The conjecture on sum-of-squares representations for varieties of minimal degree

Let $X\subset \mathbb{P}^n$ be a nondegenerate irreducible variety of minimal degree with dense real points. A quadratic form nonnegative on $X$ is a quadratic form whose restriction to $X(\mathbb{R})$ is nonnegative, and two sum-of-squares representations are considered equivalent in the sense used for such representations.

Sum-of-squares representation conjecture. A generic quadratic form nonnegative on $X$ has exactly

$$2^{\operatorname{codim}(X)}$$

inequivalent representations as a sum of $\dim(X)+1$ squares.

This conjecture extends the established count for surfaces of minimal degree to varieties of higher dimension. The preceding results establish the corresponding statement for nondegenerate irreducible surfaces of minimal degree with dense real points, while the higher-dimensional case is left open.