The generic determinant conjecture for the Werner-state matrix H

Let $d\ge3$, let $X,Y\in M_d$, and let $H(X,Y)$ be the Hermitian matrix associated with the biquadratic form used to study 2-distillability of the Werner state at $t=1/2$. Define

$$D(X,Y)=\det H(X,Y).$$

A pair $(X,Y)$ is generic in the source when $X$ and $Y$ are linearly independent and some linear combination of them is nonsingular.

Generic determinant conjecture for the Werner-state matrix $H$. $D(X,Y)\ne0$ for generic $(X,Y)$ and $d\ge3$.

The paper presents this as a simpler, stronger conjecture than the preceding positivity conjecture, and proves that it would imply that conjecture. The source does not establish the claim, so its status remains open.