Arithmetic finiteness conjecture for standard Milnor–Witt modules

Let $S=\operatorname{Spec}\mathbf Z$. For an $R$-linear MW-module $M$ over $S$, write $\big(\mathrm{FT}^{R}\big)$ for the property that, for every $S$-scheme $X$ of finite type, all integers $p,q$, and every line bundle $\mathcal L$ on $X$, the groups $A_p(X,M_q,\mathcal L)$ are finitely generated $R$-modules. The MW-modules under consideration are $\operatorname{K}^{MW}_*$, $\operatorname{K}^{M}_*$, $I^*$, $\mathcal{GW}_*$, $\mathcal W_*$ and $\mathcal K_*$. Arithmetic finiteness conjecture. The MW-modules $\operatorname{K}^{MW}_*$, $\operatorname{K}^{M}_*$, $I^*$, $\mathcal{GW}_*$, $\mathcal W_*$ and $\mathcal K_*$ over $\operatorname{Spec}\mathbf Z$ satisfy the property $\big(\mathrm{FT}^{\mathbf Z}\big)$. This is an arithmetic finiteness expectation for the usual MW-modules; the source notes that it is intended for arithmetic schemes such as number rings or finite fields, since analogous finiteness fails over general schemes—for example, $A_0(k,\operatorname{K}^{M}_1)=k^*$ for an infinite field $k$. Its resolution is not specified in the supplied text.