Unimodularity of the right transformation in a diagonal reduction

Let $A_*$ be a $G$-algebra, let $A=\operatorname{Quot}(A_*)$, and let

$$R=A[\partial;\sigma,\delta],\qquad R_*=A_*[\partial;\sigma,\delta],$$

where $R_*$ is a $G$-algebra. For a matrix $M\in R^{p\times p}$, there are square matrices $U,V,D$ with entries in $R_*$ such that

$$UMV=D,$$

where $D$ is diagonal and $U,V$ are unimodular over $R$. Right-transformation unimodularity conjecture. If $D$ has only one non-constant polynomial entry, then $V$ can be chosen to be unimodular over $R_*$. The claim formalizes the phenomenon observed in the examples: under the stated single non-constant-entry condition, the right transformation can remain unimodular over the coefficient subalgebra rather than only over its quotient field extension. No resolution is supplied in the given text.