Euler-symmetric embedding conjecture for Picard-rank-one equivariant compactifications

Let $X$ be a smooth Fano variety of Picard number one that is an equivariant compactification of a vector group. Euler-symmetric embedding conjecture. There is a suitable projective embedding under which $X$ can be realized as an Euler-symmetric projective variety. The source attributes this formulation to Fu and Hassett; it is stated as a conjecture and no resolution is given here.