In his paper "Algebraic K-Theory and classfield theory of arithmetic surfaces", Annals of Mathematics 114 (1981), Spencer Bloch proved the following result: if $A$ is a finitely generated regular $\mathbb{Z}$-algebra of Krull dimension 2 which is flat over $\mathbb{Z}$, then the K-group $K_0(A)$ is finitely generated (Corollary 4.6 of the paper). He deduces this from Corollary 4.4, and I have two questions about this reduction step.
First, to state Corollary 4.4, we need to introduce some notation. Let $k$ be a number field, $\mathcal{O}_k$ its ring of integers, $X$ a connected smooth projective curve over $k$, and $\mathcal{X}$ a regular, proper $\mathcal{O}_k$-scheme with generic fiber $\mathcal{X}_k \cong X$.
Corollary 4.4 states that, in this situation, if $X$ also has a $k$-rational point, then $K_0(\mathcal{X})$ is finitely generated.
In Remark 4.5 it is claimed that the existence of a $k$-rational point is unnecessary for this conclusion. However, it seems to me that the argument in Remark 4.5 implicitly assumes that the generic fiber $X$ is geometrically connected. Namely, in the remark it is assumed that $X(k^{\prime}) \neq \varnothing$, that is, $X$ has a $k^{\prime}$-rational point for some $k^{\prime} \supseteq k$ finite, and then it is assumed that $X_{k^{\prime}}$ has a field of fractions, hence that $X_{k^{\prime}}$ is connected. But this implies that $X_{k^{\prime}}$ and therefore $X$ are geometrically connected. This, in turn, implies that the fibers of $\mathcal{X}$ at closed points are connected, which simplifies the K-groups occurring in the localization sequence: we have
$$K_0(\mathcal{X}_{\mathbf{F}_v}) \cong \mathbb{Z} \oplus \mathbf{Pic}$$
with the notation of the paper. If the fibers are not connected, we get more copies of $\mathbb{Z}$, and it is not clear to me from the argument in Remark 4.5 how these additional copies of $\mathbb{Z}$ are annihilated by elements from $K_1(X)$ in the localization sequence.
Question 1 How can one deduce from Corollary 4.4 that $K_0(\mathcal{X})$ is finitely generated if the generic fiber $X$ is not geometrically connected?
Question 2 Can one prove that $K_0(A)$ is finitely generated (whenever $A$ is a finitely generated regular $\mathbb{Z}$-algebra of Krull dimension 2 which is flat over $\mathbb{Z}$) from the case of surfaces $\mathcal{X}$ for which the generic fiber is geometrically connected?
