In http://www.math.uiuc.edu/K-theory/0357/ Karpenko utters the following,

$Conjecture$ $1.6.$ If an anisotropic quadric $X = Q$ possesses a Rost correspondence,
then the quadratic form (defining $X$) is a minimal Pfister neighbor.

Since the Paper has been released 15 years ago, i wonder if there has been made any progress on the conjecture.

Everything regarding Rost-Projectors is defined within the paper.

I am quite new to the subject and i am "working" myself through most of the papers in order of their release,so it might be that my question is obsolete today.
On the other hand, there is also a remark in "Rost Projectors and Steenrod Operations" by Karpenko and Merkurjev from 2002 that the problem is still not solved, except for the cases of $dim(X) = 3,7$.