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subject and I am "working" myself through most of the papers in order of their release, so it might be that my question is obsolete today.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 $dim(X) = 3,7$.



Rost Correspondence and minimal Pfister-Neighbors
Last week i asked Professor Nikita Semenov and he replied that this is still an open Problem.


