The positivity conjecture for the Werner-state matrix H

Let $M_d$ denote the complex $d\times d$ matrices. For $X,Y\in M_d$, let $H(X,Y)$ be the Hermitian matrix associated in the source with the biquadratic form governing 2-distillability of the Werner state at $t=1/2$.

Positivity conjecture for the Werner-state matrix $H$. The matrix $H(X,Y)$ is positive semidefinite for all $X,Y\in M_d$.

The source states that this is an equivalent formulation of the 2-distillability conjecture. Positivity is proved there for diagonal $X$ and $Y$, while the general case remains open; numerical evidence is reported for $d=3$, and the case $d=4$ is described as open.