Let $R$ be a commutative unital ring, and $R^{m\times k}$ denote the set of $m\times k$ matrices with entries from $R$. A matrix $U\in R^{m\times m}$ is elementary if $U$ is obtained from the identity matrix $I$ by adding a $R$-multiple of its row $i$ to row $j$ for some $1\leq i\neq j\leq m$. The multiplicative group generated by the elementary matrices is denoted by $E_m(R)$.
Let $M(x,y)\in \mathbb{C}[x,y]^{m\times k}$ be such that for each $n\in \mathbb{Z}$, there are $u_n(y)\in E_m(\mathbb{C}[y])$ and $a_n,b_n\in \mathbb{C}^{m\times k}$ with $$u_n(y)\;\!M(n,y)=a_n+y\;\! b_n$$
Question: Does there exist $U(x,y)\in E_m(\mathbb{C}[x,y])$ and $A,B\in \mathbb{C}[x]^{m\times k}$ such that in $\mathbb{C}[x,y]^{m\times k}$ $$U(x,y) M(x,y)=A+y\;\!B\ ?$$
I suspect the answer might be no, but I haven't yet found a counterexample.
