Theta-pairing sign conjecture for odd-dimensional hypersurface singularities

Let $S$ be an admissible isolated hypersurface singularity of dimension $n$, and let $G(S)_\mathbb{Q}$ be the rationalized Grothendieck group on which Hochster's theta pairing $\theta$ is defined. A quadratic form is positive semi-definite if it takes nonnegative values on every vector.

Theta-pairing sign conjecture. If $n$ is odd, then the pairing

$$(-1)^{\frac{n+1}{2}}\theta$$

is positive semi-definite on $G(S)_\mathbb{Q}$.

The conjecture is proposed on the basis of a theorem proving the corresponding positivity statement over $\mathbb{C}$ and a semi-definiteness result in characteristic zero. Its validity in the stated generality remains open in the source.