Bass' stable range of $\mathbf Z[X]$

There's a comment at the top of page 993 of

L N Vaseršteĭn, A A Suslin, "Serre's problem on projective modules over polynomial rings, and algebraic K-theory", Math. USSR Izv., 1976, 10 (5), 937–1001,

to the effect that one of the authors had proved that the stable rank of $\mathbb{Z}[X]$ is 3, but unfortunately without any hint as to which or where or how, although they do show there that the stable rank of $\mathbb{Z}[X_1,\dots,X_n]$ is equal to $n+1$ if $n>1$.

After spending considerable time trying to construct a counterexample, I turned to Google and found the book "Rings Related to Stable Range Conditions" by Huanyin Chen, which claims, on page 338, that the stable range of ${\mathbb Z}[X]$ is in fact 2. This would explain my inability to find a counterexample (though not, perhaps, my willingness to put aside other urgent projects in order to look for one). However, I've not been able to understand his argument.

There is clearly a typo where he says "Clearly,${\mathbb Z}[x]$ is a euclidean integral domain; hence it is a Dedekind domain". Presumably he means ${\mathbb Z}$ instead of ${\mathbb Z}[x]$? But then I cannot fully follow the rest of the argument, possibly because it references Example 12.1.14, which is not part of the preview available on either Amazon or Google books.

In any event, though you probably already know this, the simplest class of nontrivial unimodular rows over ${\mathbb Z}[X]$ consists of those of the form $(1+aX,bX^m,cX^m)$ where $b,c$ and $m$ are arbitrary and some power of $a$ lives in the ideal $(b,c)$. I tried to find $a,b,c$ for which this row was provably not reducible, but I was insufficiently clever to pull this off.

Edit This claim that the stable range of ${\mathbb Z}[X]$ is 2 is not correct according to this other answer of mine.

Sorry for putting this in a separate answer, but I think it will be cleaner this way.

The stable range of $\mathbb{Z}[x]$ is equal to 3.

I believe I now understand Vaserstein's intended argument that it's $\ge 3$:

(1) There are rings of the form $A={\mathbb Z}[x]/(h)$ such that $SK_1(A)\neq 0$. One way to get such a ring is to start with the ring of integers in a quadratic field, let $I$ be a "sufficiently small" ideal (I confess to not being exactly sure what this means) and look at the subring generated by $1$ and $I$. For some definition of "sufficiently small", Bass has shown that this gives us $SK_1(A)\neq 0$.

(2) Take a non-zero element of $SK_1(A)$ and represent it by a Mennicke symbol $[\overline{f},\overline{g}]$.

(3) Then $(f,g,h)$ is a unimodular row over $\mathbb{Z}[x]$.

(4) Clearly, the Mennicke symbol $[\overline{f},\overline{g}]$ does not lift to $K_1({\mathbb Z}[x])$.

(5) It follows from Lemma 17.1 of the paper referenced by Jeremy Rickard that the row $(f,g,h)$ is not reducible.

I'm still just a tad unclear on why it is, in point (1), that we can take the kernel of ${\mathbb Z}[x]\rightarrow A$ to be principal. Is this obvious? I'll add a comment if I nail this down.