This does not hold in general.

An explicit counterexample is given in:
Karpenko, Nikita A. Codimension 2 cycles on Severi-Brauer varieties. K-Theory 13 (1998), no. 4, 305–330. (Reviewer: Jean-Pierre Tignol) 16K20 (14C15 19E15). (Available on the author's webpage https://sites.ualberta.ca/~karpenko/publ/ch2.pdf).

Fix a field $k$. Let $p$ be an odd prime. Let $A$ be a central division $k$-algebra of degree $p^2$, exponent $p$, and assume $A$ decomposes into a product $D_1\otimes D_2$ of two smaller algebras (both necessarily of degree $p$). Let $X$ be the Severi-Brauer variety associated to $A$. By Proposition 4.7 of the article above, the quotient $\text{Gr}^2K(X)$ contains torsion but, by Proposition 5.3 of the article above, the group $\text{CH}^2(X)$ is torsion free.

Similarly, one can show if $A$ is a division algebra of index $8$ and exponent $2$ decomposing into a product of smaller algebras then the same conclusion holds. A more general statement is true for the prime $2$ but it depends on the indices of the tensor powers of $A$.