General linear group over subring of field need not be conjugacy-closed

Statement
It is possible to have a field $$K$$, a unital subring $$R$$ of $$K$$ (note, in particular, that $$R$$ is an integral domain), and a natural number $$n$$ such that there exist matrices $$A,B \in GL_n(R)$$ such that $$A$$ and $$B$$ are conjugate in the fact about::general linear group $$GL_n(K)$$ but not in the general linear group $$GL_n(R)$$.

Related facts

 * General linear group over subfield is conjugacy-closed

Example of the integers and the rationals
Let $$R$$ be the ring of integers $$\mathbb{Z}$$ and $$K$$ be the field of rational numbers $$\mathbb{Q}$$. Consider the matrices:

$$A = \begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}, B = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$$

These matrices are conjugate in $$GL(2,\mathbb{Q})$$, by the matrix:

$$\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$$.

On the other hand, the matrices are not conjugate in $$GL(2,\mathbb{Z})$$. (Not ethat although the above matrix has integer entries, it is not in $$GL(2,\mathbb{Z})$$ because its inverse does not have integer entries).