Exactness conjecture for the homological t-groups

Let $F$ be an infinite field, let $k$ be either $\mathbb{Q}$ or $\mathbb{Z}/l$ for a prime $l$, and assume $K_2^M(F)\otimes k=0$. Let $\mathfrak{t}_r^{(k)}(F)$ denote the homological $t$-group appearing in the paper, and let $H_1(F^*,k)$ be the first homology of the multiplicative group with coefficients in $k$. Exactness conjecture for the homological t-groups. For every $n\geq 3$, the sequence

$$\mathfrak{t}_{n-2}^{(k)}(F)\longrightarrow\mathfrak{t}_{n}^{(k)}(F)\longrightarrow H_1(F^*,k)\longrightarrow 0$$

is exact. By the preceding discussion, this conjecture implies the stated parity conjecture for the primitive groups; its validity is left open in the excerpt.