Higher Serre's conjecture II for the $C^q_{\mathrm{sc}}$ property

Let $q$ be a non-negative integer and let $K$ be a field. Say that $K$ has the $C^q_{\mathrm{sc}}$ property if, for every finite extension $L/K$ and every principal homogeneous space $Z$ under a semisimple simply connected $L$-group, one has $N_q(Z/L)=\mathrm{K}^\mathrm{M}_q(L)$. Higher Serre's conjecture II. If $K$ has cohomological dimension at most $q+2$, then $K$ has the $C^q_{\mathrm{sc}}$ property. This conjecture seeks a higher-dimensional replacement for Serre's conjecture II; the supplied text does not state a resolution.