  Let $X$ be a connected component of the maximal orthogonal grassmannian of a
generic $n$-dimensional quadratic form $q$ with trivial Clifford invariant.
Consider the canonical epimorphism $\phi$ from the Chow ring of $X$ to the
associated graded ring of the coniveau filtration on the Grothendieck ring of
$X$. In \cite{Kar2018} Karpenko proved that $\phi$ is an isomorphism for all
$n\leq 12$ (conjecturally for all $n$). Recently, in \cite{Yagita} Yagita
showed that $\phi$ is not an isomorphism for $n=17, 18$. In the present paper,
together with Yagita's results for $n=17,18$, we show that the map $\phi$ is
not an isomorphism for all $13\leq n\leq 22$. In particualr, the case $n=13$
gives the smallest dimensional maximal orthogonal grassmannian whose two graded
rings are not isomorphic.
